Place chicken thighs in a resealable plastic bag. Add 1/2 cup of marsala wine and seal bag, removing as much air as possible. Refrigerate for 2 hours.
In a medium saucepan, melt butter over medium heat. Add mushrooms and cook until mushrooms are softened. Add flour and cook for 2-3 minutes or until mixture is thickened, stirring often. Add chicken broth and remaining marsala wine. Reduce heat to low and simmer until thickened, stirring often to remove lumps. Season with salt and pepper. Keep warm while grilling chicken.
Remove chicken from bag of marsala and discard liquid. Pat chicken dry. Drizzle with olive oil and Season with salt and pepper.
Grill over medium high heat for 3-4 minutes per side or until chicken is cooked through and juices run clear.
Remove from grill.
To assemble sandwiches, place chicken thighs on bun bottoms. Top each with a slice of folded cheese, a spoonful of warm sauce and a basil leaf. Finish with bun tops and serve.