Step 1 Bring a large, wide-rimmed pot of salted water to a boil. (Use about 1 tablespoon of salt for every 2 quarts of water.) Place a baking sheet inside the oven and heat the oven to 375 degrees.
Step 2 Blanch the rapini in the boiling water until just tender, about 2 minutes. Remove with tongs or a strainer and place in a colander to drain, but do not discard the cooking water. Bring the water back to a boil. When the rapini is cool enough to handle, trim it into 2-inch pieces.
Step 3Cook the potatoes in the boiling water until tender, about 10 minutes. Drain the potatoes in a colander, shaking them to break up the edges a bit. Wipe any potato bits out of the blanching pot and return it to the stove.
Step 4Carefully remove the hot baking sheet from the oven. Pour 3 tablespoons of olive oil on the sheet and mix one-half teaspoon of paprika and the garlic into the oil. Put the potatoes on the sheet, stirring to thoroughly coat. Sprinkle with 1 teaspoon salt and a little pepper. Bake 20 minutes, stirring once or twice, until the potatoes have a nice crust.
Step 5Heat the remaining olive oil in the pot on the stove over medium heat. Add the onions and saute until golden and tender, about 8 minutes. Add the remaining one-fourth teaspoon paprika; stir to mix thoroughly. Add the sausage, stir to coat with the oil, and cook 30 seconds. Add the rapini and chicken stock. Stir to mix the sausage and onions with the rapini, cook over medium heat until the rapini is tender, about 15 minutes. Add salt and pepper to taste. To serve, place 1 cup rapini mixture in the center of a warmed soup bowl. Surround with the roasted potatoes.