Heat the olive oil in a large skillet over medium heat. Add the onion and garlic and cook until softened. Add the mushrooms and season with salt and pepper. Cook until the mushrooms are tender and lightly browned. Remove from heat.
In a medium saucepan over medium heat, melt 2 tablespoons of butter. Whisk in the flour until smooth. Slowly whisk in the milk and chicken broth until combined. Cook until thickened and bubbly. Remove from heat.
Add the remaining tablespoon of butter to the skillet with the mushroom mixture and stir to combine. Pour in the sauce from the saucepan and stir to combine.
Add the cooked egg noodles to the skillet with mushroom mixture and stir to combine.