Heat the olive oil in a large skillet over medium heat. Add the garlic and cook for 30 seconds. Add the spinach and wilt, about 3 minutes; season with salt and pepper. When cool, squeeze excess water out of the spinach and chop. In a bowl, toss the spinach with the goat cheese.
In a food processor, pulse the walnuts with the 2 tablespoons flour until finely chopped; transfer to a shallow bowl. In another shallow bowl, beat the eggs with 1/4 teaspoon salt. Place the remaining 1/2 cup flour on a clean plate.
Coat 1 side of each chicken breast with the flour, then dip in the egg mixture and the walnuts. Place the chicken on a work surface, coated side down, and spread the top with one-fourth of the spinach mixture, leaving a 1/2-inch border. Roll up from the long side, jelly-roll style, and place, seam side down, on a baking sheet. Refrigerate for 20 minutes.
Preheat the oven to 350 degrees . Bake the chicken until cooked through, about 30 minutes. Let stand, covered loosely with foil, for 15 minutes; slice into rounds.