Melt the butter in a 12-inch cast-iron skillet set over medium heat. Add the bell pepper and cook until it begins to brown slightly, 5 to 6 minutes.
Add the garlic, corned beef and cabbage, thyme, oregano, and black pepper to the skillet and stir to combine. Once combined, spread the hash evenly over the pan and firmly press down with a spatula.
Place a heavy lid, heatproof plate, or second pan directly atop the hash and allow to cook until browned, approximately 10 minutes. After 10 minutes, stir mixture, press down again, top with the lid, and allow to cook another 5 to 6 minutes, or until browned. Serve immediately.