Boil the potatoes. Scrub the potatoes and place in a medium saucepan or pot. Cover with water by about 2 inches and add a big dash of kosher salt. Bring to a boil, then reduce the heat to medium (the water should be gently bubbling). Cook until fork-tender, about 20 minutes or so depending on the size of the potatoes.
Meanwhile, prepare the whipped feta. In the bowl of a food processor fitted with an S blade, add the feta, Greek yogurt, and Aleppo pepper. Close the lid and blend on low speed. While blending, use the top opening to pour in a good drizzle of olive oil (about 1 to 2 tablespoons). Continue to blend until the feta is whipped to a fluffy, smooth consistency.
Drain and cut the potatoes. When the potatoes are fully cooked through and tender, drain them and allow them to cool just briefly so you are able to handle them. Using a sharp knife, carefully make a large cross cut through each potato, but do not fully slice through the bottom. The wedges of potato should open up almost like a bloomed flower but the bottom of the potato should still be intact.
Broil. Place a rack about 6 inches from your broiler and turn the broiler on. Transfer the potatoes to 4 separate ramekins or a small dish that can hold them snugly together. Drizzle all over with a good glug of olive oil and broil for about 3 to 5 minutes, watching carefully, or until the potatoes are crisp and golden brown on the edges.
Assemble and serve. Remove the potatoes from the oven and spoon the whipped feta in the center of each potato. Zest on the lemon and sprinkle the parsley on top to finish. Serve immediately.
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Notes
Look for the larger versions of Yukon Gold potatoes if you can–they should be about the same size as standard russet potatoes.