Place potatoes into a large pot and cover with water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 30 minutes. Drain.
Let potatoes cool briefly, 5 to 10 minutes. Peel and place potatoes in a bowl and mash until smooth. Add flour and knead just until dough is smooth without overworking the potatoes. Season with salt. Cover bowl with a towel and let dough rest, about 10 minutes.
Divide dough into 4 parts. Roll each into a long rope about 1 inch in diameter. Cut each rope into 1-inch-long pieces; press into the middle of each using your thumb or a fork to shape gnocchi.
Bring a large pot of water to a boil. Add gnocchi; boil until they float to the top, 2 to 3 minutes. Transfer cooked gnocchi to a bowl or plate.
Heat a nonstick saucepan over medium heat. Add garlic; cook until lightly browned and toasted, about 2 minutes. Add tomato puree and green onion; let simmer until slightly thickened, about 5 minutes. Add cherry tomatoes. sun-dried tomatoes, paprika, oregano, basil, 3/4 teaspoon salt, and pepper. Let sauce simmer until flavors blend, about 10 minutes more. Serve with gnocchi.