In a large heavy skillet, heat 2 tablespoons of the oil over medium high heat until hot, then stir in the garlic, and cook, stirring, until golden, about 2 minutes. Stir in the broccoli, 1/4 teaspoon each salt and pepper, and cook stirring with tongs, until any liquid has evaporated, about 3 minutes. Let the broccoli mixture cool to warm.
Place each chicken breast, smooth-side-up, on a work surface. Place the palm of one hand on top of one breast, then insert a pairing knife to make a length-wise pocket in each breast. Divide the broccoli filling between the breasts, stuffing them full. Reserve any leftover spinach to serve on the side. Rub the chicken with the remaining 1 tablespoon oil, then season all over with 3/4 teaspoon salt and 1/2 teaspoon pepper.
Preheat the oven to 425°F.
Place the chicken breasts on an oiled baking sheet and bake until cooked through, about 30 minutes. Let stand 5 minutes before slicing and serving.
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