In a large pan or skillet, heat the unsalted butter and your choice of oil (olive, coconut, or avocado) over medium-high heat. Allow the fats to melt and combine, making sure the pan is evenly coated.
Add the diced onion to the hot butter and oil mixture. Sauté for about 3 minutes, stirring occasionally, until the onion becomes softened and slightly translucent.
Add the roughly chopped green cabbage to the pan with the sautéed onion from Step 2. Cook for 4-5 minutes, stirring occasionally, until the cabbage is softened and its edges begin to caramelize. I like to let the cabbage sit for a minute or two without stirring so it can get extra golden on the edges.
Stir in the thinly sliced Lacinato kale, finely chopped garlic, coarse salt, ground black pepper, and red pepper flakes. Mix well to incorporate all the ingredients, then cook for 8-10 minutes, stirring occasionally, until the kale is wilted and both the cabbage and kale are nicely browned and softened.
Pour in the vegetable broth to deglaze the pan, using a wooden spoon to scrape up all the browned bits from the bottom. Let the mixture cook for about 2 minutes, allowing the broth to reduce slightly. Continue cooking the vegetables for another 30 seconds until the aroma is deep and fragrant.
Taste the vegetables and adjust the seasoning with extra pepper if desired. Sprinkle the chopped fresh parsley over the top and serve immediately while hot. For extra brightness, I sometimes squeeze a little fresh lemon juice just before serving.