Melt butter in a large skillet over medium-high heat. Saute carrots and onion for about 4 minutes. Add garlic during the last few seconds so that it becomes fragrant but does not burn.
Combine broccoli and chicken broth in a microwave-safe bowl. Steam in the microwave until tender, about 3 minutes. Drain off chicken broth. Coarsely chop the florets and add to the carrot mixture. Stir to combine and set aside.
Remove skin from the rotisserie chicken. Using a sharp knife, carefully separate pieces. Cut meat into small cubes, making sure to pick through and discard any bones.
Combine butter and flour in a heavy saucepan over medium heat. Stir to form a paste. Slowly pour in milk. Cook, whisking constantly, until sauce thickens, about 7 minutes. Stir in white Cheddar cheese and garlic salt.
Preheat the oven to 350 degrees F (175 degrees C).
Set aside 1/2 cup Cheddar cheese for the topping. Spread about 1/2 cup of the sauce over the bottom of an 8x8-inch casserole dish. Add a layer of lasagna noodles, broccoli mixture, cheese, and chicken. Repeat layers, ending with sauce. Sprinkle reserved Cheddar cheese and panko bread crumbs over the top.
Cover lasagna with foil and bake in the preheated oven for 35 minutes. Uncover and bake for 5 minutes more. Allow to rest for 7 minutes before serving.