Heat oven to 400° F. In a small bowl, combine the butter and chopped garlic.
On a work surface, dividing evenly, top the prosciutto slices with the chicken and garlic butter; season with ½ teaspoon salt and ¼ teaspoon pepper. Roll up the chicken in the prosciutto; drizzle with 1 tablespoon of the oil. Place on a rimmed baking sheet and roast until cooked through, 20 to 25 minutes.
Meanwhile, heat the remaining tablespoon of oil in a large skillet over medium heat. Add the sliced garlic and cook, stirring, until fragrant, 1 to 2 minutes. Add the spinach and ¼ teaspoon each salt and pepper and cook, tossing, until the spinach begins to wilt, 1 to 2 minutes more. Serve with the chicken.