Place the drained sun-dried tomatoes and 1/2 cup (120 ml) water in a blender or food processor and blend until smooth, about the consistency of sour cream; set aside.
Slice the chicken into 1/2-inch-thick pieces and season both sides with salt and pepper.
Heat 1 tbsp olive oil (use oil from the sun-dried tomato jar if using) in a large nonstick skillet over medium-high heat; add the chicken and cook 3–4 minutes per side until cooked through and golden.
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) until al dente; drain, briefly rinse if desired, and set aside.
In the same large skillet, combine the sun-dried tomato puree, red pepper flakes, minced garlic, half & half, and grated Parmesan; bring to a gentle simmer over medium heat and cook 1–2 minutes to meld flavors.
Add the chopped chicken and drained penne to the sauce and toss to coat evenly.
Sprinkle shredded mozzarella evenly over the pasta mixture and place the skillet (or transfer to an oven-safe baking dish) under the broiler for about 5 minutes, watching closely, until the cheese is melted and lightly browned.
Garnish with fresh thyme and serve immediately.