Pour milk into a medium pot and bring to a simmer over medium-high heat. Slowly add cornmeal while whisking.
Let polenta simmer, stirring often, until it begins to spit & pop. Lower heat and continue cooking, stirring often and scraping the bottom, until polenta thickens and pulls away from the side of the post, about 30 minutes.
While polenta is cooking, fry the diced bacon in a large skillet over high heat until not quite crispy, about 5 minutes. Remove the bacon from the skillet and drain all but 1 T. of bacon grease.
Sauté the mushrooms in bacon grease until al dente, about 5 minutes. Return bacon to the skillet and lower heat to keep warm until polenta is ready.
When polenta has thickened, stir in goat cheese, butter, and salt until smooth. Divide between two bowls.
Whisk egg, parmesan, thyme, and pepper in a medium bowl. Add hot mushrooms & bacon and stir quickly to heat the egg and create a silky sauce.
Top polenta with carbonara and serve immediately with extra parmesan & pepper.
If polenta thickens too much or begins to set, whisk in hot water one tablespoon at a time until smooth.