In large skillet, melt the butter and whisk in the flour- cook, stirring, until browned. Slowly whisk in the broth, stirring constantly. Cook until slightly thickened. Add the vegetables and simmer for 20 minutes. Stir in the cheese until melted. Add the milk and the beer and stir to combine. Add the parsley and cook on low heat for another 5 minutes.
Season to taste with salt and pepper. Serve hot.
Notes
Substitute vegetable broth for chicken broth to make this vegetarian.