Spread ¼ cup flour in shallow dish. Pat chicken dry with paper towels and season with salt and pepper. Working with 1 breast at a time, dredge breasts in flour.

Heat 1 tablespoon oil in 10-inch skillet over medium-high heat until just smoking. Lay chicken in skillet and cook until well browned on first side, 6 to 8 minutes. Flip chicken, reduce heat to medium, and continue to cook until chicken registers 160 degrees, 6 to 8 minutes; transfer to serving platter and tent loosely with aluminum foil.

Heat remaining 1 tablespoon oil in now-empty skillet over medium heat until shimmering. Add shallot and cook until softened, about 2 minutes. Stir in remaining ½ teaspoon flour and cook for 1 minute. Whisk in broth and wine, scraping up any browned bits. Bring to simmer and cook until sauce is slightly thickened, about 5 minutes. Stir in any accumulated chicken juices and simmer for 30 seconds. Off heat, whisk in butter and parsley and season with salt and pepper to taste. Pour sauce over chicken and serve.