Melt butter in a pot over medium-high heat. Add mushrooms and onions and cook until softened, about 4-5 minutes.
Mix flour, salt, pepper, and 1½ cups broth in a bowl. Add to pot and stir to combine.
Add 1½ cups broth to the pot. Bring to a boil and stir until thickened, about 2 minutes.
Reduce heat to low and stir in half and half. Simmer for 15 minutes, stirring occasionally.
