Heat the olive oil in a thick-bottomed saucepan.
Fry two-thirds of the prosciutto for one minute.
Melt 50g of butter in the pan.
Add the wine and simmer for 3 minutes.
Mix egg yolks, parmesan, and pecorino in a bowl.
Season the cheese mixture with salt and pepper.
Cook the tagliatelle until al dente.
Drain the pasta while reserving some cooking water.
Combine the pasta with the prosciutto mixture.
Stir in the egg mixture using residual heat.
Thin the sauce with reserved water if necessary.
Stir in remaining prosciutto and butter.
Serve in warm bowls with extra parmesan.