Season chicken on both sides with salt, black pepper, and cayenne and set aside.

Melt 3 tablespoons butter in a skillet over medium heat; add sage leaves and fry until crisp, 1 to 2 minutes. Remove and reserve.

Add chicken breasts to the same skillet, and cook until just cooked through, about 4 minutes per side. Remove to a plate and cover loosely with foil.

Add shallots to the skillet; sauté for 1 minute. Pour in cider and vinegar, and bring to a simmer over medium-high heat. Cook until reduced by half, which will only take a few minutes.

Reduce heat to low and place chicken back in, along with any accumulated juices from the plate. Cook for a few minutes, turning occasionally, and basting with the hot sauce until the chicken is hot. An instant read thermometer inserted into the center should read 165 degrees F (74 degrees C). Remove to a serving plate.

Add remaining 1 tablespoon butter to the pan, and turn off the heat. Swirl pan until butter emulsifies into the sauce. Taste for seasoning and serve on chicken, garnished with fried sage.