Wash the green beans, trim the ends, and cut the potatoes into even pieces. Slice the sausage and set everything aside.
Heat olive oil in a large skillet over medium heat. Add sausage and brown well on both sides.
Add onion and bell pepper. Cook until soft, then stir in garlic and cook for 30 seconds.
Add butter and let it melt. Sprinkle in flour if using, and stir for 1 minute.
Pour in chicken broth and add seasonings. Scrape the pan to release flavor.
Add green beans and potatoes. Stir gently to coat.
Cover and simmer on low for 25–35 minutes until potatoes are fork-tender.
(For oven method: transfer to a baking dish, cover, and bake at 375–400°F for 35–45 minutes.)
Taste and adjust seasoning. Let rest 5 minutes before serving.