Preheat the oven to 350°F (175°C).
Bring a large pot of salted water to a boil. Add the chopped cabbage and cook for 5 minutes until slightly tender. Drain and set aside.
In a large skillet, melt 3 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. Remove from heat.
In a large mixing bowl, combine the cooked cabbage, sautéed onions and garlic, shredded cheddar cheese, shredded mozzarella cheese, sour cream, mayonnaise, whole milk, salt, black pepper, smoked paprika, and dried thyme. Mix thoroughly until well combined.
Transfer the cabbage mixture into a greased 9×13-inch casserole dish and spread evenly.
In a small bowl, mix the crushed crackers with the melted butter until the crumbs are evenly coated.
Sprinkle the cracker topping evenly over the cabbage mixture.
Bake in the preheated oven for 35 to 40 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it cool for 5 minutes before serving.