Preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper.
In a medium skillet over medium heat, add 1 tablespoon of the 2 tablespoons extra-virgin olive oil. Add the minced garlic and sauté about 1 minute, until fragrant and turning lightly golden.
Add the coarsely chopped baby spinach to the skillet and sauté about 5 minutes, until wilted. Season the spinach with 1/2 teaspoon kosher or sea salt and 1/2 teaspoon pepper. Remove from heat and let the spinach mixture cool completely.
When the spinach mixture has cooled, stir in 1/2 cup grated Parmesan cheese and 1 cup shredded mozzarella until evenly combined. Set the filling aside.
Prepare a shallow dish or plate with 1 cup Italian-style breadcrumbs (or panko). Put the remaining 1 tablespoon extra-virgin olive oil on a second plate or in a small bowl for brushing/coating.
Place each boneless, skinless chicken breast between sheets of plastic wrap or in a zip-top bag and use a meat tenderizer to pound each breast until thin enough to roll easily. Cut each pounded breast in half lengthwise to make smaller cutlets (total should be 8 cutlets).
Pat each cutlet dry with paper towels. Lay one cutlet flat, place a portion of the spinach-cheese filling along one short end, and tightly roll the chicken over the filling, ending with the seam side down. Repeat with remaining cutlets.
Lightly brush or drizzle the outside of each roll with the reserved olive oil, then roll each one in the breadcrumbs, pressing gently so crumbs adhere. Place the coated rolls seam-side down on the prepared baking sheet.
Bake in the preheated oven for 30 minutes, or until the rolls are golden and the chicken is cooked through (an internal temperature of 165°F/74°C). Let the rolls rest 3–5 minutes before serving.