Start by bringing a large pot of salted water to a boil. Add the orecchiette pasta and cook according to package instructions until al dente, about 10–12 minutes. Drain and set aside.
In a large skillet over medium heat, melt the butter and add the sliced mushrooms. Sauté undisturbed for about 3–4 minutes until browned, then add the minced garlic and cook for another 3–4 minutes until caramelized.
In a small bowl, whisk together cornstarch and half-and-half until smooth. Pour into the skillet with milk, salt, pepper, thyme, and Parmesan cheese. Simmer over medium-low heat for about 6–8 minutes until the sauce thickens.
Add the drained orecchiette pasta to the skillet and toss gently until well coated with the sauce.