Fill a 12-inch high-sided skillet with 2 inches of water, cover and bring to a boil. Trim 1/4 inch from the stem ends of the broccoli rabe (as they can be dried out) and discard.
Add 1 tablespoon salt and the trimmed broccoli rabe to the boiling water. Boil until the stems are tender when pierced with a knife, 4 to 5 minutes. Remove with tongs to a colander to drain. When cool enough to handle, press out as much excess water as possible from the broccoli rabe, and then chop into 1- to 2-inch pieces. Reserve 1/2 cup of the cooking water.
Quarter the potatoes and slice the garlic into thin slivers.
Heat the olive oil in a large skillet over medium-high heat. When the oil is hot, add the potatoes, cut-side down, and cook until browned, about 4 minutes. Turn with tongs or a slotted spoon to the other cut side and cook until browned, 3 to 4 minutes more.
Lower the heat to medium, add the garlic and stir until it softens a bit, about 1 minute. Stir in the paprika until the potatoes are coated, about 30 seconds more. Season with 1/2 teaspoon salt and a few grinds of pepper. Add the reserved cooking water and bring to a gentle simmer. Cover and cook the potatoes until tender when pierced with a knife, about 5 more minutes. If there is still a lot of water in the skillet, uncover, increase the heat and reduce until only 2 to 3 tablespoons remain, enough to coat the broccoli rabe. Add the chopped broccoli rabe to the skillet and toss to coat. Drizzle with a generous amount of olive oil and serve.
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