In small microwavable bowl, mix oil and garlic. Microwave on High 1 minute; set aside. Place flour on plate; stir in salt and pepper. Coat chicken with flour, shaking off excess.
In 12-inch nonstick skillet (1 1/2 inches deep), place 1 tablespoon of the heated oil without garlic pieces. Heat oil over medium-high heat 1 to 2 minutes or until hot but not smoking. Add chicken, cutting large pieces in half, if necessary, so all chicken fits in skillet; cook 4 to 6 minutes, turning once, until no longer pink in center and golden brown. Remove chicken from skillet; place on plate and cover to keep warm.
In same skillet, cook mushrooms over medium-high heat 1 minute, stirring occasionally, until thoroughly heated. Stir in wine and soup with heatproof rubber spatula or wooden spoon to scrape up brown bits from bottom of skillet. Cook 5 to 7 minutes, stirring occasionally, until thoroughly heated.
Meanwhile, set oven control to broil. Split rolls; place cut side up on large cookie sheet. Broil 6 to 8 inches from heat 1 to 2 minutes or until toasted and golden brown. Brush cut sides with remaining oil mixture with garlic pieces; top evenly with fontina cheese. Broil 30 to 60 seconds or until cheese is melted.
Return chicken to skillet. Reduce heat to medium-low; simmer uncovered 2 to 3 minutes, turning chicken occasionally, until chicken is coated with sauce. Divide chicken evenly among rolls. With slotted spoon, divide mushrooms and onions over chicken. Pour wine sauce into 4 (4-oz) ramekins or dipping bowls. Sprinkle parsley on individual plates; place sandwiches and ramekins of sauce for dipping on plates.