Lets say I have a transaction on ETH chain and one on BSC chain, I start the transaction on ETH chain and it asks for my signature. What I want to do is have it just ask for my signature once and have the transaction on the other chain automatically get executed without asking me for the signature again.
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This is why we have chainId. Basically whenever you sign a transaction on an ethereum based blockchain, the chainId is included in that signature. BSC chainId is 56, and eth mainnet is 1. So if you tried to broadcast a signed transaction from ethereum to bsc, the nodes on bsc would reject it because the chainId inside the signed data is wrong.
So in essence, no, I do not believe this is possible. And you wouldn't want to be able to do it, because without it every time you made a transaction on one EVM chain, exploiters could replay it on every other chain whether you wanted to make that transaction on that chain or not.
If you wanted to use a users signature to allow execution of a transaction on two chain though, you could use smart contracts to check a signature and allow some action instead. The only caveat there is that someone else needs to start the transaction on each blockchain, so you need to provide the incentive for someone to spend that gas, or manage with a centralized transaction starter. There's probably tools out there for this exact thing.
Also, some bridges allow message passing across chains, which could be used to keep a transaction flow going across two blockchains.
