Stir together first 3 ingredients in a shallow bowl. Dredge chicken in flour mixture; place on a wire rack. Let stand 15 minutes. Dredge chicken in flour mixture again; return to rack.
Melt butter with olive oil in a large skillet over medium heat. Cook chicken, in batches, 5 minutes on each side or until golden brown. Remove chicken to a plate.
Add shallots to skillet; cook, stirring often, 2 minutes or until golden brown. Add mushrooms and garlic, and cook, stirring often, 10 minutes or until mushrooms are tender. Stir in Champagne and thyme; bring to a boil, stirring to loosen browned particles from bottom of skillet. Reduce heat, and return chicken to skillet. Cover and simmer 10 minutes or until done.
Transfer chicken to a serving platter. Stir cream into mushroom mixture. Cook 5 to 6 minutes or until thickened. Add salt and pepper to taste. Serve sauce immediately over chicken.