Remove the sausage casing if necessary, and cut into chunks or slices. Cut the mushrooms into chunks or slices of similar sizes.
Heat the oil in a large pot. You can do it in a pan too, but if you choose a pot big enough to cook the pasta, you only dirty one dish. Saute the mushrooms for a few minutes, then add the sausage. Make sure they're fully cooked. Set aside.
In the same pot, remove excess oil and boil water. Cook the linguine until al dente or your preferred doneness - we're not judging here!
While the pasta cooks, put the egg yolks in a mixing bowl. Add milk, 1/2 cup Parmesan, lemon juice and salt & pepper. Whisk until combined.
Drain the linguine, saving 1 cup of water. Put the pasta, sausage and mushroom back in the pot and mix together.
Pour the egg sauce over top and stir well to make sure everything gets coated. The sauce should thicken but not get chunky. If it's too thick, add pasta water a little at a time until you've reached your desired consistency.
Serve individual portions, then sprinkle with remaining cheese and parsley/chives.