VW Jetta stalls out on start

I have a 2001 Jetta with 105,000 miles on it. whenever i start it, the engine turns over, and then about 3 seconds later will stall out unless you give it some gas. this is much more pronounced in cold weather, but also happens in the warmer months as well. once you give it gas, and get it going, everything else works fine. i thought it might be a bad fuel pump, but i had it checked and they said it was working fine.

does anyone know what the problem might be? is it worth fixing? if i don’t fix it, will it cause further damage down the line?

thanks all!