Add the butter and garlic to a deep skillet and sauté over medium heat for one minute. Add the sliced mushrooms, salt, and pepper, and continue to sauté until the mushrooms have softened, all of their moisture has evaporated from the skillet, and the edges are beginning to brown.
Add the fettuccine to the skillet along with the vegetable broth and stir to combine. It's okay if the broth doesn't fully submerge the pasta.
Place a lid on the skillet, turn the heat up to medium-high, and allow the broth to come up to a boil. When it reaches a boil, give the pasta a quick stir, replace the lid, then turn the heat down to medium-low.
Continue to let the pasta simmer in the broth for about 10 minutes, stirring occasionally (always replacing the lid), or until the pasta is tender. There should be a little saucy liquid left in the bottom of the skillet.
Add the heavy cream to the skillet and stir to combine. Turn the heat off then add the Parmesan and continue to stir the pasta until the Parmesan is melted. Give the pasta a taste and add salt or pepper if needed. Serve immediately.
*Fettuccine is my preferred pasta for this recipe, but you can try any other pasta shape you like. Some readers have had success with linguine, penne, and bow-tie pasta.**Heavy cream creates a super-rich and luscious sauce. Because it's so thick, it also helps the Parmesan to melt without clumping. Milk, which has a much higher water content, will produce a thinner sauce, and you may have difficulty getting the Parmesan to melt in instead of clumping. One way I've gotten around this in the past is to use a combination of milk and cream cheese. The cream cheese both emulsifies the Parmesan and thickens the sauce (see how this technique is used in myCreamy Sun-Dried Tomato Pasta).***I use a 3-quart deep skillet for this pasta, but you can use any pot or Dutch oven that has a wide bottom and a lid. Make sure to use a burner on your stovetop that is similar in size to the bottom of your pot or skillet to ensure even heating.