Heat 4 tablespoons of butter in a pan on medium heat. Saute onions until softened, about 8 minutes, stirring occasionally. Add garlic and thyme, and cook for an additional 2 minutes while constantly stirring.
Stir in chopped mushrooms and any fragments from the sliced mushrooms (the best slices of the baby bella mushrooms are reserved for later use). Stir in thoroughly to coat with the contents of the pan. Stir occasionally for 20 minutes, allowing some moisture to evaporate from the mushrooms.
Add stock to the pan and lower heat to medium-low. Cook for half an hour uncovered.
Pour contents of the pan into a blender and puree until is achieves desired consistency. Return to the pan and add milk. Season with salt and pepper, adjusting to taste.