Heat a large skillet, Dutch oven, or heavy-bottomed pot over medium heat. Add the olive oil or bacon grease, followed by the smoked sausage. Cook for 4 to 5 minutes, turning as needed, until the sausage is browned on both sides. Transfer the sausage to a plate and set aside.
Add the butter and diced onion to the same pot. Cook for 3 to 4 minutes, stirring occasionally, until the onion softens. Add the garlic and cook for about 30 seconds, just until fragrant.
Add the potatoes, smoked paprika, garlic powder, onion powder, black pepper, Cajun seasoning, and red pepper flakes. Stir until the potatoes are evenly coated in the seasonings.
For fresh green beans, add them now with the chicken broth, Worcestershire sauce, and optional sugar. Stir and bring the liquid to a gentle simmer. Cover, reduce the heat to medium-low, and cook for 25 to 35 minutes, stirring occasionally, until the potatoes are fork-tender and the beans are tender. Add a splash of broth if the pot becomes dry before the potatoes finish cooking.
If using canned green beans, add the broth, Worcestershire sauce, and optional sugar to the seasoned potatoes first. Cover and simmer for 15 minutes, then stir in the drained green beans. Cover and continue cooking until the potatoes are fork-tender, about 10 to 15 minutes longer.
Return the browned sausage and any collected juices to the pot. Simmer uncovered for 8 to 10 minutes, stirring occasionally, until the sausage is hot and the broth has reduced enough to lightly coat the ingredients.
Taste and add salt as needed. Stir in the apple cider vinegar or lemon juice, if using. Serve hot.