Heat the butter and oil in a large skillet until the butter is melted; stir to mix the oil and butter. Add the cabbage and stir until it is coated. Cover, turn heat to low, and simmer until the cabbage is soft, about 8-10 minutes. Remove the cover, turn up the heat, and stir for 2-3 minutes until most of the liquid in the pan has evaporated.
Heat the milk until scalding (a microwave will do this in about 2 minutes). Melt the remaining butter in a small saucepan. Add the flour and whisk for 1-2 minutes until a paste forms. Pour in the hot milk, turn down the heat, and continue whisking until you have a thin sauce. Add the cheese and stir until it is melted.
Beat the eggs in a small bowl. Add 1 tablespoon of the white sauce and mix. Then add the eggs to the rest of the sauce. If you are using herbs or spices, mix them in now.
Grease a 9x12 inch loaf pan. Preheat oven to 375 F.
Fold the cabbage mixture into the sauce and mix well. Transfer this mixture to greased loaf pan. Sprinkle the breadcrumbs on top. Bake in the preheated oven for 35-45 minutes, until the edges of the cabbage begin to brown.
Allow to cool for 10-15 minutes, then slice and serve warm.