Pat 1 ½ pounds thin-sliced boneless, skinless chicken breasts dry with paper towels. Season all over with 1 teaspoon of the kosher salt and ½ teaspoon black pepper.

Heat 1 tablespoon of the olive oil in a large (12-inch) high-sided skillet over medium-high heat until shimmering. Add half the chicken pieces in a single layer and sear until browned, 1 to 2 minutes per side. Transfer to a large plate (chicken will not be cooked through). Add the remaining 1 tablespoon olive oil to the pan and repeat browning the remaining chicken. Transfer to the same plate.

Reduce the heat to medium and melt 3 tablespoons unsalted butter in the skillet. Add 1 pound thinly sliced cremini mushrooms, 3 minced garlic cloves, and the remaining 1 teaspoon kosher salt. Cook, stirring occasionally, until tender and any liquid from the mushrooms has evaporated, about 8 minutes. Add 1 tablespoon plus 1 teaspoon all-purpose flour, stir to evenly coat the mushrooms, and cook for 1 minute.

Stir in 1 ½ cups low-sodium chicken broth and ¼ cup heavy cream until no lumps from the flour remain. Bring to a boil. Reduce the heat to maintain a simmer. Stir in ¼ cup finely grated Parmesan.

Return the chicken and accumulated juices to the pan and pile 5 ounces baby spinach on top. Cover and simmer until the sauce thickens slightly, the chicken is cooked through, and the spinach wilts, 2 to 3 minutes. Stir to incorporate the spinach into the sauce. Taste the sauce and season with more kosher salt as needed.