Step 1: In a small bowl, combine 3/4 teaspoons each of the salt and pepper and the paprika. Sprinkle the mixture over the skin-side of the chicken.
Step 2: Melt the butter in a large enameled and lidded pan (like a Staub dutch oven). Over medium high heat. When sizzling, add two of the chicken thighs, skin side down. Cook until the skin is golden brown, about 6 minutes. Turn chicken over and cook for another two minutes. Remove to a plate. Repeat with the two remaining pieces.
Step 3: While the chicken browns, peel the potatoes. Cut in half, lengthwise. Then cut into 1 1/2 inch pieces.
Step 4: When chicken is out of the pan, add the carrots to the pan and cook, stirring occasionally, until slightly browned - about 7 minutes. Add the chopped garlic and cook for an additional minute, stirring frequently so it doesn't burn.
Step 5: With the carrots still in the pan, add the flour. Cook and stir for a minute. The flour should not brown. Add the broth and stir until the roux is incorporated. Add the remaining 1/4 teaspoon salt and pepper, along with the thyme sprigs. Taste for seasoning and adjust if necessary.
Step 6: Place the browned chicken, skin side up, atop the carrots. Partially cover the pan with the lid, leaving a small opening for steam to escape. Reduce heat to medium low and cook until the chicken is done, about 20 minutes depending on the thickness of the thighs.
Step 7: To serve, remove the thyme sprigs. Serve over mashed potatoes.
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