Season the flour with salt and pepper, then mix together in a shallow bowl. Lightly dust both sides of the chicken with the flour, shaking off any excess.
Add the olive oil to a skillet and heat over medium-high heat. When the oil is hot, add the chicken skin-side down and cook until the skin is golden brown, 5 to 6 minutes. Flip the chicken and cook on the other side until golden brown, about 4 more minutes. Remove the chicken from the skillet and set aside.
Add the mushrooms, shallot and fennel to the skillet and cook just until beginning to soften, about 5 minutes. Sprinkle with 1 teaspoon of the excess dredging flour and stir well. Add the white wine, chicken broth, garlic, rosemary, thyme and lemon juice, then bring to a simmer, scraping the bottom of the skillet. Add the chicken back to the skillet skin-side up and place in the oven. Cook until the chicken is cooked through and the vegetables are tender, about 20 minutes.
Serve with Pistachio Pesto Pasta if desired and garnish with basil leaves, lemon zest and lemon slices.
Pistachio Pesto Pasta:
Bring a large pot of salted water to a boil. Add the pasta and cook according to the package directions.
While the pasta is cooking, make the pistachio pesto. Combine the pistachios, basil, olive oil, Parmesan, garlic, lemon zest and juice and some salt and pepper in a food processor and process until well combined.
Reserve 1 cup of the pasta water and drain the pasta. Toss the pasta with the pistachio sauce, thinning with the pasta water for the desired consistency if necessary. Serve with more Parmesan and basil leaves.