Soak skewers in water for at least 30 minutes and up to 2 hours.
Meanwhile, in a small bowl, whisk oil, garlic, lemon zest, lemon juice, Italian seasoning, red pepper flakes, 1 tsp. salt, and a few grinds of black pepper.
In a large bowl, toss chicken, bouillon (if using), and ⅓ cup sauce. Add breadcrumbs, Parmesan, a large pinch of salt, and several grinds of black pepper and gently toss to coat.
Thread 4 to 5 chicken pieces onto each skewer. Top chicken with any remaining breadcrumbs, gently pressing to adhere.
Prepare a grill for medium-high heat; preheat for 5 minutes (or heat a grill pan over medium-high heat).
Grill chicken, turning occasionally, until golden brown and an instant-read thermometer inserted into thickest part registers 165°, about 5 minutes per side. Transfer skewers to a plate or platter.
Add parsley to remaining sauce. Spoon some reserved sauce over top. Sprinkle with more Parmesan. Serve with remaining sauce alongside.
