Preheat the oven to 350°F (175°C). Grease a 9×13-inch baking dish with 1 tablespoon of butter.
Bring a large pot of salted water to a boil. Add the chopped cabbage and cook for 5 minutes until slightly tender. Drain well and set aside.
In a large skillet, melt 2 tablespoons of butter over medium heat. Add the chopped onion and cook for 5 minutes until softened.
Add the minced garlic to the skillet and cook for 1 minute until fragrant.
Sprinkle the flour over the onions and garlic and stir constantly for 2 minutes to form a roux.
Gradually whisk in the milk, continuing to stir until the sauce thickens, about 5 minutes.
Remove the skillet from heat and stir in salt, black pepper, smoked paprika, 1 1/2 cups shredded cheddar cheese, and 1/4 cup Parmesan cheese until melted and smooth.
Add the drained cabbage to the cheese sauce and stir to combine evenly.
Transfer the cabbage mixture to the prepared baking dish and spread evenly.
In a small bowl, combine the crushed crackers, remaining 1/2 cup cheddar cheese, remaining 1/4 cup Parmesan cheese, and 1 tablespoon melted butter. Sprinkle this mixture evenly over the casserole.
Bake uncovered for 35-40 minutes, until the topping is golden brown and the casserole is bubbly.
Remove from the oven and let cool for 5 minutes. Garnish with chopped fresh parsley before serving.