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1
Melt 2 tablespoons butter with vegetable oil in heavy large skillet over medium-high heat.
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2
Add all onions; cook until deep brown, stirring often, about 8 minutes.
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3
Add sugar and saute until sugar dissolves, about 1 minute.
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4
Add 1/2 cup vinegar; saute until almost all liquid evaporates, about 1 minute.
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5
Transfer to small bowl.
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6
Preheat oven to 200F.
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7
Mix salt, 1/2 teaspoon cumin, and pepper in small bowl.
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8
Rub chicken breasts all over with spice mixture.
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9
Heat 2 teaspoons olive oil in heavy large skillet over medium-high heat; add chicken breasts, skin side down.
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10
Saute until cooked through, about 4 minutes per side.
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11
Transfer chicken to baking sheet.
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12
Keep warm in oven.
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13
Heat remaining 2 teaspoons olive oil in same skillet over medium heat.
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14
Add garlic and saute 20 seconds.
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15
Add caramelized onions, 1/4 cup vinegar, tomato, olives, capers, and Sherry.
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16
Cook until liquid reduces slightly, stirring constantly, about 2 minutes.
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17
Stir in 1 1/2 teaspoons cumin, chicken broth, lime juice, sage, thyme, and rosemary.
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18
Add avocado; cook until sauce thickens, stirring constantly, about 3 minutes.
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19
Reduce heat to low.
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20
Add 4 tablespoons butter and stir until blended into sauce.
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21
Season with salt and pepper.
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22
Place 1 chicken breast on each of 6 plates; spoon sauce over.