Melt 2 tablespoons butter in a large Dutch oven over medium heat. Add diced onion and carrots. Cook 5 minutes until onion is translucent.
Add minced garlic and cook 30 seconds until fragrant.
Push vegetables to one side. Add remaining 1 tablespoon butter to empty space. Once melted, sprinkle in flour and whisk into butter. Cook 2 minutes, stirring constantly to form roux.
Gradually pour in chicken broth while whisking constantly. Add Italian seasoning, garlic powder, onion powder, smoked paprika, salt, and pepper. Bring to gentle simmer, whisking until slightly thickened.
Add broken lasagna noodles to pot. Reduce heat to medium-low and simmer 10 minutes, stirring every few minutes to prevent sticking.
Stir in broccoli florets. Cook 5-7 minutes until broccoli is tender-crisp and noodles are cooked through.
If adding chicken, stir it in now along with milk. Heat through 3-4 minutes without boiling.
Turn heat to low. Gradually stir in shredded cheddar cheese in batches, stirring until each addition is fully melted before adding more.
Taste and adjust seasonings. Serve immediately garnished with fresh parsley and extra cheese if desired.