Preheat oven to 450 degrees. In a skillet over medium heat, add 1/4 cup olive oil, heat slowly. Turn down oil to low-medium and add the garlic; fry, stirring, until golden. With a slotted spoon, transfer the garlic to a plate lined with a paper towel to absorby excess oil. Set the garlic aside to cool and reserve the oil for the potatoes.
Place the fingerling potatoes in a pot, cover with cold water, add about 1 tbsp of salt and bring to a simmer. cook until a fork pierces through easily, about 20 minutes. Drain and set aside to cool.
Trim the ends of the rabe and wash. Pat dry and spread out on a large sheet pan. drizzle 2 tbsp of the olive oil over each bunch of rabe and toss to coat evenly. Place on a rack in the upper third of the oven. (One bunch will fill one sheet pan. You can roast more than one bunch at a time on two racks, if you have more pans.) roast 12-15 minutes or until tender. Remove from the oven and drizzle 2 tbsp of the sherry vinegar over each bunch and toss.
Slice the potatoes into inch-thick coins and fry in one layer in the reserved garlic oil until golden. Flip the potatoes and repeat. Transfer the potatoes to a large bowl and, while still hot, sprinkle with the smoked paprika and a large pinch of sea salt. Toss till evenly coated.
To serve: Place the rabe on individual plates or a large platter, surround with the potato coins. Sprinkle with a little more salt and finish with a drizzle of extra virgin olive oil. Garnish with the olives and fried garlic.