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Step 1
On a sheet pan, season the chicken all over with salt and pepper and set aside. Heat 2 tablespoons of olive oil and 1 tablespoon butter in a large skillet (12 inches in diameter) over high heat. Once the oil-butter mixture is melted and bubbling, sear chicken in two batches. Lay half of the chicken down, spaced about 1 inch apart, and cook, undisturbed, until golden on the bottom but not cooked through, about 7 minutes per batch. Transfer thighs back to sheet pan, seared side up. Preheat the oven to 425º F.
Step 2
Turn down the heat to medium and heat the remaining 2 tablespoons olive oil and 2 tablespoons butter. Once melted and bubbly, add the mushrooms, tossing them in the oil-butter and then let cook undisturbed until they lightly brown. Toss again and keep cooking, tossing frequently, until they soften, have more browning and taste hot and juicy in the center, about 10 minutes. Transfer the mushrooms to the chicken sheet pan.
Step 3
Deglaze the skillet with the white wine, scraping up the browned bits with a wooden spoon. Add the onion, garlic, rosemary and a big pinch of salt, and cook, stirring regularly until softened, about 5 to 7 minutes. Pour in the tomato, stir to combine with the onion, warm the tomato then taste, season with salt as needed, then tuck the chicken and mushrooms into the tomato. Bring to a simmer, place the skillet in the oven and cook until the sauce has thickened slightly and the chicken is tender, about 30 minutes. Let cool for about 15 minutes before serving.
Step 4
Once ready to serve, divide 1½ cups of the arugula on four serving plates, then tuck the remaining ¼ cup arugula into the tomato sauce. Spoon the sauce, chicken and mushrooms over the plated arugula.