Heat oil in large nonstick skillet over low heat. Add shallots; cook and stir 6 to 8 minutes or until caramelized.
Stir in wine, broth and thyme. Cook over medium-high heat 8 to 10 minutes or until liquid is reduced to 1/2 cup.
Combine butter and flour; whisk into sauce. Stir in 1/4 teaspoon salt and 1/4 teaspoon pepper. Cover; keep warm.
Combine ground beef, 2 tablespoons parsley, 1/8 teaspoon salt and 1/8 teaspoon pepper in large bowl, mixing lightly, but thoroughly. Lightly shape into four 1/2- inch thick patties. Set aside.
Place mushrooms on grid over medium, ash-covered coals; grill, uncovered, 16 to 18 minutes or until tender, turning occasionally. About 10 minutes before mushrooms are done, move mushrooms to outer edge of grid.
Place patties on center of the grid; grill 8 to 10 minutes (over medium heat on preheated gas grill, covered, 7 to 9 minutes) until instant-read thermometer inserted horizontally into center registers 160 degrees F, turning occasionally.
Place bread slices on grid; grill until toasted, turning once.
Reheat sauce, if necessary. Spread 1/2 of cheese on toasted bread slices. Top each with lettuce leaf, mushroom and burger; drizzle evenly with sauce. Crumble remaining goat cheese over tops; sprinkle with parsley, as desired.