Sauté half of chopped onion in 1 tablespoon hot oil in a medium skillet over medium-high heat until tender. Add garlic, and sauté 30 seconds. Remove from heat, and let cool.
Combine onion mixture, beef, next 4 ingredients, 1/4 teaspoon salt, and 1/2 teaspoon pepper. Shape into 6 patties.
Cook patties in remaining 2 tablespoons hot oil over medium-high heat 3 minutes on each side or until browned. (Do not cook until done.) Remove patties and set aside.
Add remaining half of chopped onion to drippings in skillet, and sauté over medium heat until tender; add mushrooms, and sauté 3 minutes. Whisk in flour, and cook, stirring constantly, 1 minute; whisk in wine, broth, and remaining 3/4 teaspoon salt and 1/2 teaspoon pepper. Bring to a boil; reduce heat to low and simmer, stirring occasionally, 5 minutes.
Place patties in a lightly greased baking dish; top evenly with gravy.
Bake, covered, at 350° for 25 minutes or until done.
Note: To reheat, bake, covered, at 350° for 45 minutes or until bubbly.