Heat oven to 425° F. Toss the turnips, carrots, 2 tablespoons of the oil, and 1/4 teaspoon each salt and pepper on a rimmed baking sheet. Roast, tossing once, until tender, 14 to 16 minutes.
Meanwhile, heat 2 tablespoons of the oil in a large skillet over medium heat. Season the chicken with the oregano and 1/4 teaspoon each salt and pepper. Cook until an instant-read thermometer inserted into the thickest breast registers 165° F, 6 to 8 minutes per side.
Combine the anchovies, garlic, the remaining 4 tablespoons of oil, and 1/4 teaspoon each salt and pepper in a small pot. Cook, stirring, over medium heat, until the anchovies are broken down and the garlic is fragrant, 1 to 2 minutes. Stir in the lemon juice.
Serve the chicken and vegetables topped with the anchovy oil and chives.
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