Position a rack in the center of the oven and preheat to 350 degrees .
Butterfly the chicken by slicing horizontally into the thick, long side of each breast, taking care not to slice all the way through. Open each breast like a book and place on a work surface. Using a mallet, pound the chicken to even it out a bit. Season both sides with salt and pepper.
In a medium skillet, heat 1 tbsp. EVOO, one turn of the pan, over medium. Add the garlic and stir 30 seconds. Add the spinach and stir until wilted, 3 to 4 minutes. Add the nutmeg; season. Drain the spinach well, let cool, then finely chop. (You will have 3 to 4 tablespoons chopped spinach.) In a medium bowl, mix the spinach with the cheeses; season.
Spread some of the spinach mixture on one side of each chicken breast, leaving a small border. Close the other half of the chicken breast over the filling to cover. Wrap each chicken breast with 2 slices prosciutto.
In a large, ovenproof skillet, heat the remaining 2 tbsp. EVOO, two turns of the pan, over medium-high. Add the chicken and cook until the prosciutto is browned, 2 to 3 minutes on each side. Transfer the skillet to the oven and roast until the chicken is cooked through, about 12 minutes. Halfway through cooking, add the lemon halves, cut side down, to the pan and cook until the lemons are browned.
Divide the chicken breasts among plates. Squeeze the juice from the lemon halves over the chicken.