"Before the client’s signed my contract, should I start working on that big design project?"
The simple answer: No.
The more complex answer: No. Every minute that you bill will vanish in a whiff of non-billable smoke if the client decides to back out at the eleventh hour. For some designers, it only takes one client walking away from a contract to encourage strong boundaries. Others get burned over and over again, which can be crippling for your cashflow.
As part of your pre-contact negotiations, let the client know that you won't start work until your contract is signed. Every extra day that they take to sign the contract, you should tack on that extra time to the end of your schedule, instead of trying to make up for it. This should be clear to the client, in writing, as part of your terms and conditions.
If you haven’t extended them credit, you shouldn’t work until they’ve provided you with your first payment as well.
If they say the check’s in the mail, you let them know that the project will start when you’ve received it (and cashed it). If it's "held up in accounting," then this gives them an incentive to go breathe fire down said accountant's neck.
If you have worked with your client a number of times in the past, and they have paid you promptly for services rendered, you may consider extending them credit and getting started… but only after you have a signed contract.
If you're tempted to use a letter of intent (LOI) to negotiate a big fancy contract or master services agreement, know that it's essentially tantamount to extending credit if you don't get paid up front for your time.
And if you've set up your contract so your payments are gating the work, make it clear to the client well before you get to the end of each phase that you'll be expecting payment to proceed. Otherwise, they may be quite pissed that the project is going to be behind schedule because you hadn't communicated an essential responsibility to them.