FOR THE SPINACH: Adjust oven rack to middle position; heat oven to 200 degrees. Heat 2 tablespoons of oil and garlic in 12-inch skillet over medium-high heat; cook, shaking pan back and forth, until garlic begins to sizzle and is light golden brown, about 3 minutes. Add spinach by the handful, using tongs to stir and coat spinach with oil.

Once all spinach is added, sprinkle 1/4 teaspoon salt, pepper flakes, and lemon zest over top and continue stirring with tongs until spinach is uniformly wilted and glossy green, about 2 minutes. Using tongs, transfer the spinach to colander set in sink and gently squeeze spinach with tongs to release excess juices. Return the spinach skillet and sprinkle with lemon juice and stir to coat. Transfer spinach to large platter and drizzle with remaining 1 teaspoon olive oil and sprinkle with additional salt to taste. Cover loosely with foil and transfer to oven to keep warm. Wipe out skillet with paper towels.

FOR THE CHICKEN: Season both sides of each cutlet with salt and pepper. Heat 2 teaspoons oil in now empty 12-inch skillet over medium-high heat until smoking; place 4 cutlets in skillet and cook without moving them until browned, about 2 minutes. Using a spatula, flip cutlets and continue to cook until second sides are opaque, 15 to 20 seconds. Transfer cutlets to large plate. Repeat with remaining 4 cutlets and 2 teaspoons oil. Tent plate loosely with foil and transfer to oven to keep warm while making sauce.

FOR THE SAUCE: Return the now-empty skillet to medium-high heat and add 1 tablespoon butter; when melted add figs and cook, stirring occasionally, 2 minutes. Add water and any accumulated juices from chicken and cook until figs are plump and water has evaporated, about 1 minute. Add vinegar, sugar, thyme sprigs, and bay leaves and simmer, stirring occasionally, until the vinegar is just barely covering the bottom of the pan, 7 to 8 minutes. Remove the skillet from the heat, discard the herbs, and slowly whisk in the remaining 2 tablespoons butter. Season to taste with salt and pepper.

Shingle chicken cutlets over spinach. Spoon balsamic sauce over chicken and serve.