All reagents are symmetric, so how do you design a reaction where you produce more left or right? So starting from purely symmetric reagents and catalysts, how do you produce asymmetry?

When the chirality of the product is an autocatalyst to its own chirality. So if your reaction starts off with a little more left than right due to molecular randomness, the left is an autocatalyst which means there is a runaway process and you end up at the end of the reaction with a lot more left than right.

If you repeat the reaction from scratch, it is random if you end up with a left product or a right product. But the product of your reaction is not a 50-50 mix left-right! I.e. you created asymmetry in your solution from purely symmetric solvents.