Drain most of the oil from the sun-dried tomatoes, reserving about 1 tablespoon if desired; place the tomatoes and 1/2 cup water in a blender or food processor and puree until smooth and creamy.
Slice the chicken breasts into 1/2-inch-thick slices and season both sides with salt and pepper.
Heat 1 tablespoon olive oil (or reserved sun-dried tomato oil) in a large nonstick skillet over medium-high heat and cook the chicken slices 3–4 minutes per side, until cooked through and no longer pink.
Remove the chicken to a cutting board, cut into bite-sized pieces, and set aside.
Bring a large pot of lightly salted water to a boil and cook the Chickapea penne according to package directions, about 3–5 minutes; drain, briefly rinse if desired, and set aside.
In a large skillet over medium heat, combine the sun-dried tomato puree, red pepper flakes, minced garlic, half & half, and grated Parmesan; bring to a gentle simmer and cook for about 1 minute, stirring until the sauce is smooth.
Add the cut chicken and cooked penne to the sauce and stir until everything is evenly coated.
Sprinkle shredded mozzarella evenly over the top, transfer to an oven-safe dish or place the skillet under a broiler, and broil about 5 minutes until the cheese is melted and bubbly—watch carefully to avoid burning.
Garnish with fresh thyme and serve immediately.