Start a large pot of water boiling for the pasta.
In a large skillet, melt 1 tablespoon of the butter over medium heat. Add roughly two cloves of minced garlic to the pan, along with the mushrooms. Cook for approximately 5 minutes, stirring occasionally, or until the largest pieces of mushrooms are soft. Remove the mushrooms from the pan and set aside.
Melt the other tablespoon of butter in the pan. Add the flour and stir to make a paste.
Continuing to stir the butter/flour paste ( roux! ), add the remaining minced garlic, salt, and pepper. Stir over medium heat for one minute.
Slowly stir in the mushroom broth, ½ a cup at a time. Make sure all of the broth is absorbed into the roux before adding the next ½ cup of broth. Add the wine after all of the broth has been added.
(You should multitask here -- now's a good time to add the pasta to the boiling water, and cook according to package directions. My guess is 3 - 5 minutes. )
Cook the sauce in the skillet for 5 minutes, until the broth and wine have reduced by ¼. This isn't a science, so don't take out the measuring stick. Just make sure that the sauce has reduced enough to have a thicker, sauce-like consistency instead of a watery consistency. If it gets too thick, you can add some water to the pan to thin it back out!
Add the cooked mushrooms, parsley, cheese, and fresh thyme to the pan. Stir to combine.
Drain the pasta, and add to the pan with the mushroom sauce. Toss to coat.
Serve with a sprinkle of Pecorino Romano cheese on top.