Preheat the oven to 175 degrees. Cut each chicken breast in half horizontally to form 2 thin cutlets; place between 2 sheets of plastic wrap. Using a meat mallet, pound to 1/4 inch thick. Season with salt and pepper. Place the flour on a plate; dredge the chicken, shaking off excess. Top each cutlet with 1 sage leaf, then wrap in 1 slice of prosciutto, tucking the ends under the cut side of the chicken.
In a large skillet, heat 1 tbsp. EVOO, one turn of the pan, over medium-high heat. Add 1 tbsp. butter; when it stops foaming, add half the cutlets, cut side up. Cool until the prosciutto is crispy and golden, 3 to 4 minutes per side. Transfer the cutlets to a baking sheet and keep warm in the oven. Add the remaining 1 tbsp. butter to the pan and repeat with remaining cutlets.
Add the remaining 1 tbsp. EVOO to the skillet, one turn of the pan. Add the garlic and stir until aromatic, about 1 minute. Add the spinach; stir until wilted, 2 to 3 minutes. Add the nutmeg and season. Douse the greens with the lemon juice. Divide the cutlets among plates and serve with the spinach.