Prepare the following, adding each to the same medium bowl as you complete it: Dice 1 large yellow onion and 4 large celery stalks. Smash and peel 3 garlic cloves.
Heat 3 tablespoons of the olive or vegetable oil in a large pot or Dutch oven over medium heat until shimmering. Add the onion mixture, 1/4 teaspoon of the kosher salt, and 1/4 teaspoon black pepper. Cook, stirring often, until the vegetables are softened but not browned, about 8 minutes.
Meanwhile, scrub and peel 2 pounds Yukon Gold potatoes, saving the skins. Cut the potatoes into 1-inch chunks. Drain and rinse 2 (about 15-ounce) cans white beans.
Add the potatoes, white beans, 4 cups water or unsalted vegetable broth, 1 (about 13-ounce) can full-fat coconut milk, the remaining 2 teaspoons kosher salt, 1/2 teaspoon red pepper flakes if using, and the remaining 1/2 teaspoon black pepper to the pot. Increase the heat to medium high and bring to a simmer. Reduce the heat to medium and cook, stirring occasionally, until the potatoes are very soft, about 35 minutes.
Meanwhile, heat the remaining 1/2 cup olive or vegetable oil in a small saucepan over medium heat until shimmering. While the oil is heating, coarsely chop the reserved potato peels to the size of almonds. Line a plate with paper towels.
Add half of the potato peels to the oil and fry, swirling the pan occasionally, until crisp and puffed, 8 to 15 minutes. Use a slotted spoon to transfer the potato peels to the paper towels. Season with a pinch of kosher salt. Repeat frying the remaining potato peels. Reserve any leftover oil for another use.
When the soup is ready, turn off the heat. Add 1/4 cup distilled white vinegar and stir to combine. Use an immersion blender to blend the soup directly in the pot to your desired consistency (it can be totally smooth or left a bit chunky). Alternatively, transfer the soup in batches to a stand blender and blend to your desired consistency.
Taste & season with more kosher salt and vinegar as needed. Serve topped with crispy peels & chopped dill.
