Heat the oven to 450°F with the rack in the middle position. While the oven is heating up, pat the chicken dry with paper towels. Sprinkle kosher salt liberally (about 2 teaspoons) on both sides of the chicken pieces.
Heat a 12-inch oven-safe heavy bottomed skillet over medium high heat. When the pan is hot, swirl in the ghee.
Add the chicken pieces skin-side down in the hot pan. Sear, undisturbed, until nicely browned and crispy, about 4 to 6 minutes. While the skin is crisping, add some freshly ground black pepper to the meaty side.
When the skin is golden brown, flip the pieces over and cook for 2 minutes more. Then, transfer the chicken to a platter. (Note: The chicken is not ready to eat at this point. You need to finish cooking it in the oven.)
Turn the heat off and add the shallots to the empty pan. It’s plenty hot and you don’t want to burn the shallots. Cook, stirring occasionally, until slightly softened, about 1 minute.
Turn the heat to medium and toss in the summer squash and a sprinkle of salt. Saute, stirring occasionally, until the squash is lightly browned and tender, about 3 to 5 minutes.
Stir in the cherry tomatoes and garlic cloves. Cook until garlic is fragrant and some of the tomatoes have burst, about 1 minute.
Pour in the broth and balsamic vinegar and stir well to combine. Add the thyme sprigs.
Bring to a simmer and taste and adjust sauce for seasoning with more salt if needed.
Tuck in the chicken pieces, skin-side up, making sure the skin stays above the broth.
Transfer the pan to the hot oven and roast the chicken for 15-20 minutes. The dish is ready when the internal temperature of the chicken reaches at least 165°F (aim for 150°F to 155°F for chicken breast) and the vegetables have softened into a chunky sauce.
Remove the skillet from the oven. Discard the thyme sprigs and garnish with fresh basil. Serve immediately!