1. Preheat oven to 375 degrees F (190 degrees C).
2. Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally until cooked through but firm to the bite, about 7 minutes. Drain and set aside.
3. Melt butter in a large pot over medium heat. Whisk in flour, dry mustard, and black pepper; cook until smooth, about 2 minutes. Whisk in milk, a little at a time, and bring white sauce to a boil. Cook for 1 minute, whisking constantly, to make a smooth, thick sauce. Whisk 3/4 cup marble cheese into sauce until melted.
4. Stir spaghetti, broccoli, carrot and kolbassa sausage into the sauce and transfer to a 3-quart casserole dish.5. Bake in the preheated oven until casserole is heated through, about 20 minutes.