Preheat the oven to 375 degrees. In a large skillet, heat the butter until melted and slightly frothy. Add the onion and saute until fragrant and beginning to turn translucent, about 2-3 minutes.
Add the cabbage & combine with onions. Season with salt. Cook for about 5-6 minutes or until the cabbage is softened and lightly golden in spots. Stir in the garlic and cook for about a minute. Remove from heat and let the mixture cool slightly.
Add the frozen hash browns and 1 ½ cups of cheese and mix until well-combined. Smooth the top with a spatula. Bake in the preheated oven for about 30-35 minutes or until golden brown.
Add the remaining cheese to the top and bake for another 4-5 minutes until the cheese is melted and bubbly. Remove from oven and top with chopped green onion before serving.