This is like hunter's style chicken (or chicken cacciatore), minus the tomato. The sweetness of the peppers and the acidity of the wine come to the fore, creating a dish that is at once light and warming. Serve this wine-braised chicken with peppers and herbs with brown rice or lightly dressed fregola.
Sprinkle the chicken with 1½ tsp. salt and 1 tsp. pepper. Set the flour in a shallow bowl. Dredge the chicken in the flour, shaking to remove any excess. Set a large, heavy based Dutch oven (preferably 7.5 quart or larger) over medium-high heat for 1 minute. Add 2 Tbs. oil, and once it's shimmering hot, add half of the chicken, skin side down and evenly spaced. Cook, undisturbed, until the chicken browns around the edges and easily releases when you lift an edge, 2 to 3 minutes. Flip, brown the other side for 2 minutes, and then transfer to a large plate. Reduce the heat to medium, add another 1 Tbs. olive oil, and brown the remaining chicken the same way and transferring to the plate.
Add the remaining 1 Tbs. olive oil to the pot, then the peppers and onions, sprinkle with ½ tsp. salt and, still over medium heat, cook, stirring occasionally, until the onion becomes tender and translucent, about 6 minutes. Stir in the tomato paste and then the white wine, raise the heat to high, and cook, scraping the bottom of the pot with a wooden spoon, until the liquid almost completely cooks off, 2 to 3 minutes.
Add the chicken broth and thyme and bring to a boil. Reduce the heat to medium low, return the chicken to the pot, cover, and cook until the chicken cooks through and is tender, about 30 minutes. Skim any fat off the top. Season the broth with salt and pepper to taste.
Spoon the broth and chicken into shallow serving bowls, sprinkle with the Parmigiano and more thyme, and serve.