Aztec Chicken

6 tablespoons (3/4 stick) chilled butter
3 tablespoons vegetable oil
1 large red onion, halved lengthwise, thinly sliced crosswise
1/2 yellow onion, thinly sliced
1 1/2 tablespoons sugar
3/4 cup Sherry wine vinegar

1 tablespoon coarse kosher salt
2 teaspoons ground cumin
1/4 teaspoon ground black pepper
6 boneless chicken breast halves with skin
4 teaspoons olive oil

4 garlic cloves, thinly sliced
1 large tomato, peeled, seeded, diced
1/4 cup coarsely chopped pitted oil-cured black olives
1/4 cup drained capers, rinsed
1/4 cup Sherry
1/3 cup low-salt chicken broth
1/4 cup fresh lime juice
1 tablespoon chopped fresh sage
1 tablespoon chopped fresh thyme
1 teaspoon minced fresh rosemary
1 ripe avocado, halved, pitted, peeled, cut into 1/2-inch cubes
Preparation
Melt 2 tablespoons butter with vegetable oil in heavy large skillet over medium-high heat. Add all onions; cook until deep brown, stirring often, about 8 minutes. Add sugar and sauté until sugar dissolves, about 1 minute. Add 1/2 cup vinegar; sauté until almost all liquid evaporates, about 1 minute. Transfer to small bowl.

Preheat oven to 200°F. Mix salt, 1/2 teaspoon cumin, and pepper in small bowl. Rub chicken breasts all over with spice mixture. Heat 2 teaspoons olive oil in heavy large skillet over medium-high heat; add chicken breasts, skin side down. Sauté until cooked through, about 4 minutes per side. Transfer chicken to baking sheet. Keep warm in oven.

Heat remaining 2 teaspoons olive oil in same skillet over medium heat. Add garlic and sauté 20 seconds. Add caramelized onions, 1/4 cup vinegar, tomato, olives, capers, and Sherry. Cook until liquid reduces slightly, stirring constantly, about 2 minutes. Stir in 1 1/2 teaspoons cumin, chicken broth, lime juice, sage, thyme, and rosemary. Add avocado; cook until sauce thickens, stirring constantly, about 3 minutes. Reduce heat to low. Add 4 tablespoons butter and stir until blended into sauce. Season with salt and pepper. Place 1 chicken breast on each of 6 plates; spoon sauce over.