Adjust the oven rack to the lower-middle position and preheat the oven to 325°F. Pat the lamb pieces dry with paper towels and season with 3/4 teaspoon salt and pepper. Heat 1 tablespoon of vegetable oil in a Dutch oven over medium-high heat until shimmering. Brown half the lamb on all sides, about 8–10 minutes, and transfer to a bowl. Repeat with another tablespoon of oil and the remaining lamb.
Add onions, the remaining 1 tablespoon of oil, and 3/4 teaspoon salt to the Dutch oven. Cook over medium heat until softened, 5–7 minutes. Stir in garlic and rosemary, cooking until fragrant, about 30 seconds. Sprinkle in the flour, stirring constantly, and cook for 1 minute.
Slowly whisk in the white wine and 1 cup of water, scraping up any browned bits. Add the remaining 3/4 cup of water, canned tomatoes with their juices, and the browned lamb with its accumulated juices. Stir to combine. Cover and transfer the pot to the oven. Cook for 1 hour.
Remove the pot from the oven. Stir in the potatoes and sprinkle the green beans on top. Cover and return to the oven, cooking until the lamb and vegetables are tender, about 1 hour more. Adjust the stew’s consistency with extra hot water if necessary. Stir in the basil and season with salt and pepper to taste. Serve warm.