Pat 6 bone-in, skin-on chicken thighs dry with paper towels and season all over with ¾ teaspoon of the kosher salt and ¼ teaspoon black pepper.

Heat 1 tablespoon of the olive oil in a large 12-inch skillet over medium heat until shimmering. Add the chicken skin-side down and sear until the skin is golden-brown, 10 to 13 minutes. Flip and cook until the second side is golden-brown, 10 to 12 minutes more.

Transfer the chicken to a large plate skin-side up (it will not be cooked through).

Add 8 ounces thinly sliced mushrooms and ¼ teaspoon of the kosher salt to the skillet. Cook, stirring occasionally, until the mushrooms are browned, 6 to 8 minutes. Stir in 2 minced garlic cloves and 1 tablespoon all-purpose flour, and cook until the garlic is fragrant, about 30 seconds.

Stir in 1 cup half-and-half, ½ cup low-sodium chicken broth, and the remaining ¼ teaspoon kosher salt. Scrape up any browned bits from the bottom of the skillet and bring to a simmer. Reduce the heat to medium-low.

Return the chicken skin-side up to the skillet and pour in any accumulated juices from the plate. Simmer until the sauce is thick enough to coat the back of a spoon and the chicken is cooked through, 12 to 15 minutes.

Using tongs, transfer the chicken to a serving platter. Stir 3 ounces baby spinach into the sauce a few handfuls at a time. Cook until just wilted, 1 to 2 minutes. Stir in 2 tablespoons balsamic vinegar. Taste and season with more kosher salt as needed. Spoon the sauce, mushrooms, and spinach around the chicken.