Snap the stem ends of the green beans, or cut them off in a big bunch with a knife if you'd prefer.
Melt the bacon grease in a skillet over medium-low heat. Add the garlic and onions and cook for a minute. Then add the green beans and cook until the beans turn bright green, about a minute. Add the chicken broth, chopped red pepper, salt and pepper to taste. Turn the heat to low and cover the skillet with a lid, leaving the lid cracked to allow steam to escape. Cook until the liquid evaporates and the beans are fairly soft, yet still a bit crisp, 20 to 30 minutes. You can add more chicken broth during the cooking process, but don't be afraid to let it all cook away so the onions and peppers can start to caramelize.