Saute onion in 2 tablespoons hot olive oil in a large nonstick skillet over medium-high heat 8 to 10 minutes or until tender. Add walnuts and garlic; cook, stirring constantly, 2 minutes. Stir in tomato paste, and cook, stirring often, 5 minutes.
Add brown sugar and next 5 ingredients. Reduce heat, and simmer, stirring often, 30 minutes or until thickened; cool slightly.
Pulse mixture in a blender or food processor 3 or 4 times or until smooth; keep warm.
Place chicken between 2 sheets of heavy-duty plastic wrap, and flatten to 1/4-inch thickness using a meat mallet or rolling pin.
Toss together apple, ham, and thyme. Spoon 1 heaping tablespoon of apple mixture onto each chicken breast; roll up, securing with wooden picks.
Heat remaining 2 tablespoons olive oil in a large nonstick skillet over medium heat; add chicken, and cook 2 minutes on each side. Reduce heat; cover and cook 7 to 10 minutes or until done. Drizzle with sauce. Garnish, if desired.