Melt 3 tablespoons butter in a saucepan over medium heat until a pinch of flour sprinkled into the oil will just begin to bubble. Whisk in flour, stirring constantly until mixture becomes paste-like and light golden brown, about 5 minutes. Slowly add milk, stirring to avoid lumps. Continue stirring until a smooth, thick sauce is obtained, about 10 minutes more. Whisk in bouillon, paprika, and thyme. Remove from heat and let sauce cool.
Preheat the oven to 350 degrees F (175 degrees C). Lightly butter a 9×13-inch casserole dish.
Melt remaining 3 tablespoons butter in a skillet over medium-high heat. Add zucchini and onion; saute until softened, 12 to 15 minutes. Add carrot and continue cooking for 2 minutes. Remove from heat.
Add egg to the cooled sauce; whisk rapidly until smooth. Add Cheddar cheese and mix to combine.
Layer zucchini mixture evenly into the prepared casserole dish. Pour sauce over vegetables and give it a slight stir to moisten throughout. Sprinkle bread crumbs on top and spray lightly with cooking spray.
Bake in the preheated oven until bread crumbs start to brown, 20 to 25 minutes. Broil until panko is nicely browned, 2 to 4 minutes more. Remove from oven and let sit for 10 to 15 minutes before serving.