<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0202</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2008-03-14</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000724606</issuerCik>
        <issuerName>PANERA BREAD CO</issuerName>
        <issuerTradingSymbol>PNRA</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001244907</rptOwnerCik>
            <rptOwnerName>BORLAND MARK</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>6710 CLAYTON ROAD</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>RICHMOND HEIGHTS</rptOwnerCity>
            <rptOwnerState>MO</rptOwnerState>
            <rptOwnerZipCode>63117</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>SVP,Chief Supply Chain Officer</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Choice Award (Right to Buy Class A Common Stock)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <footnoteId id="F1"/>
                <footnoteId id="F2"/>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2008-03-14</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>0</value>
                    <footnoteId id="F1"/>
                    <footnoteId id="F2"/>
                </transactionShares>
                <transactionPricePerShare>
                    <footnoteId id="F1"/>
                    <footnoteId id="F2"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <value>2008-03-14</value>
            </exerciseDate>
            <expirationDate>
                <value>2008-03-28</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Class A Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <footnoteId id="F1"/>
                    <footnoteId id="F2"/>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>0</value>
                    <footnoteId id="F1"/>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Represents a choice award under the 2006 Stock Incentive Plan (&quot;2006 Plan&quot;), permitting the Reporting Person to elect to receive, effective March 28, 2008, unless otherwise determined by the Compensation and Stock Option Committee of the Board of Directors (the &quot;Compensation Committee&quot;), an award in the form of (i) up to a number of shares of restricted Class A Common Stock, par value $0.0001 per share (&quot;Class A Common Stock&quot;), as has a fair market value of $100,000 on March 28, 2008 as determined by the closing price of shares of Class A Common Stock on The Nasdaq Global Market on that date, or (ii) stock options to purchase up to a number of shares of Class A Common Stock as would cause the stock option to have a fair market value of $100,000 on March 28, 2008 based on the Black-Scholes option pricing model, or (iii) a combination of restricted stock and stock options (in 25% increments).</footnote>
        <footnote id="F2">The options would have an exercise price equal to the closing price of the Class A Common Stock on The Nasdaq Global Market on March 28, 2008, unless otherwise determined by the Compensation Committee, and vest over a three (3) year period from the date of grant, with 33 1/3% vesting annually.  The options covered by the agreement will terminate six (6) years from the date of the agreement, but will be subject to earlier termination as provided in the award agreement or the 2006 Plan.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Louis DiPietro, Attorney in Fact for Mark A. Borland</signatureName>
        <signatureDate>2008-03-18</signatureDate>
    </ownerSignature>
</ownershipDocument>
