Melt butter in a skillet; add mushrooms and garlic, and sauté until tender. Remove mushrooms with a slotted spoon, reserving drippings in skillet.
Whisk flour into reserved drippings until smooth. Cook, whisking constantly, until lightly browned. Gradually add wine and broth; cook over medium heat, whisking constantly, until thickened and bubbly. Stir in salt and pepper. Add mushrooms; cook, stirring constantly, until sauce is thoroughly heated.