1. Place each breast half between 2 pieces of plastic wrap. Pound lightly with the flat side of a meat mallet to 1/4-inch thickness.* Remove plastic wrap. Sprinkle lightly with salt and pepper; set aside.
2. In a medium skillet heat 2 teaspoons of the olive oil over medium heat; add garlic. Cook and stir about 30 seconds or until golden brown. Remove from heat. Stir in well-drained spinach, 1/3 cup of the currants, and half of the pine nuts. Season lightly with salt and pepper.
3. Divide spinach mixture among chicken breasts. Starting from a long side, roll up each chicken breast into a spiral, folding in ends.
4. Lightly grease a 3-quart rectangular baking dish. Place chicken roll-ups, seam sides down, in prepared baking dish. Brush with remaining olive oil. Sprinkle with additional pepper. (Cover and refrigerate for up to 4 hours.) Bake, uncovered, in 375 degrees oven about 25 minutes or until no longer pink and juices run clear.
5. Meanwhile, for sauce, in a small saucepan gradually stir chicken broth and apple juice into cornstarch. Add remaining 1/3 cup currants and the dried thyme, if using. Cook and stir until thickened and bubbly. Cook and stir for 2 minutes more. Remove from heat. (Cover and refrigerate for up to 4 hours. Reheat before serving.) Stir in remaining pine nuts, the butter, and the fresh thyme, if using.
6. To serve, diagonally slice each chicken breast into 1/2-inch slices. Arrange chicken slices on serving plates; drizzle with a little sauce. Pass remaining sauce.