Step 1
Preheat the oven to 375°.
Step 2
Pat dry one 4-lb. chicken, cut into 8 pieces with a paper towel and season them all over with 1 tsp. kosher salt and ½ tsp. freshly ground pepper.
Step 3
In a 12″ ovenproof skillet, heat 2 Tbsp. extra virgin olive oil over medium heat. Add half the chicken pieces, skin-side down, and cook undisturbed until the skin is golden brown and the chicken releases fairly easily from the pan, 6 to 8 minutes. (Use a splatter screen if you have one.) Flip and cook until browned on the other side, about 3 minutes. Transfer to a plate and repeat with the remaining pieces of chicken.
Step 4
Drain and discard all but a thin layer of fat, about ¼ cup, from the skillet. Add 1 large yellow onion, halved through the stem and sliced ¼” thick to the pan along with ½ tsp. kosher salt and a few grinds of pepper. Cook, tossing often, until the onion is lightly browned, 8 to 10 minutes. Add 12 oz. seedless green grapes, in small clusters on the stem (about 2 ½ cups), 4 garlic cloves, minced, and 2 tsp. fresh thyme leaves and cook for 30 seconds, being careful not to let the garlic burn.
Step 5
Off the heat, add ½ cup Champagne or prosecco and scrape any browned bits from the pan. Return the chicken pieces, skin-side up, along with any accumulated juices to the pan, nestling the larger pieces into the clusters of grapes. Transfer the skillet to the oven and bake until an instant-read thermometer inserted into the thickest part of the breast meat registers 165°, 20 to 25 minutes.
Step 6
Carefully remove the skillet from the oven and transfer the chicken to a plate to rest while you make the sauce. Add 1 tsp. Dijon mustard to the pan juices in the skillet and bring to a simmer over medium heat. Cook for 3 minutes, stirring often, until the liquid has slightly reduced. Stir in 2 Tbsp. heavy cream and cook for 1 more minute.
Step 7
Spoon the sauce onto the bottom of a large serving platter with raised edges. Thinly slice the chicken breast crosswise and arrange on the platter with the rest of the chicken pieces. Scatter the grapes around the chicken. Sprinkle with salt and black pepper, garnish with a few fresh thyme leaves, and serve.
