Heat 1 tablespoon olive oil in a large, heavy pot over medium heat; cook and stir onions, pepperoni sausage, and garlic until garlic is golden brown, 8 to 10 minutes. Transfer pepperoni mixture to a bowl and set aside.
Heat 3 tablespoons olive oil in same pot; cook and stir carrots, cabbage, kosher salt, and black pepper until coated in hot oil. Place tomatoes, zucchini, and green beans into pot and scatter onion-pepperoni mixture over the top. Partially cover the pot and simmer until vegetables are tender, about 50 minutes.
Place potatoes into a large pot and cover with lightly-salted water. Bring to a boil, reduce heat to medium-low, and simmer until tender, about 20 minutes. Drain and let potatoes cool. When cool enough to handle, cut potatoes into 1 1/2-inch pieces.
Gently fold potatoes into the vegetable stew and cook until heated through, about 3 more minutes; drizzle stew with 1 tablespoon olive oil and serve.
