Season 2 large skinless, boneless chicken breast halves (7 oz. to 8 oz. each) with salt and pepper. Coat with 2 Tbsp. flour; shake off excess.
In a large skillet, heat 1 Tbsp. olive oil over medium-high heat. Brown chicken, turning once, about 10 minutes. Transfer chicken to a plate and cover with foil.
Add 1 chopped shallot to skillet and cook on medium heat, stirring, for 1 minute. Pour in 1/3 cup dry white wine and boil for 2 minutes. Add 1 cup reduced-sodium chicken broth and bring to a boil. Add 2/3 cup fresh blueberries and a pinch of sugar and simmer until reduced by half, about 3 minutes. Remove from heat and stir in 1 Tbsp. unsalted butter; season with salt and pepper.
Arrange chicken on plates. Top with sauce.