1. On a shallow dish, combine flour, curry powder, salt, and cayenne. Dip chicken into seasoned flour to coat lightly. In a large frying pan, melt 1 1/2 tablespoons of butter. Add half of chicken and cook over medium-high heat, turning once, until white throughout, about 5 minutes. Remove from pan. Repeat with another 1 1/2 tablespoons butter and remaining chicken. Remove second batch of chicken from pan.
2. Melt remaining 1 tablespoon butter in pan. Add apple slices, reduce heat to medium, and cook, stirring, until softened and lightly glazed, about 3 minutes. Add cider and broth, stirring up browned bits clinging to bottom of pan. Simmer until sauce is lightly reduced and thickened, about 2 minutes.
3. Stir in chutney. Return chicken to pan to heat through, about 1 minute. Serve at once.