Prepare the ketchup. Combine all the ingredients except the black pepper in a small, heavy-bottomed, nonreactive saucepan. Over medium heat, bring the mixture to a bubble through the center, then reduce the heat to low. Simmer the ketchup gently, uncovered, until it has thickened and the flavors have blended well, about 10 minutes. Stir in a generous amount of black pepper and set aside. You should have about 2 cups ketchup. The ketchup tastes best when either served tepid (warm to the touch) or at room temperature.

Prepare the potatoes. Fill a large bowl with ice water. Scrub the potatoes and either peel them or dry the skins thoroughly. Cut the potatoes into 1/8-inch slices using a handheld crinkle cutter or a sharp chef’s knife, and immediately place the slices in the ice water to prevent them from oxidizing (turning brown) and also to remove some of their surface starch.

If using an electric deep-fat fryer, pour in the oil to the designated line and turn to the highest setting (or follow the manufacturer’s instructions). Otherwise, fill a 4- to 8-quart heavy-bottomed saucepan half-way with oil. Attach a deep-frying thermometer securely onto the side of the pan (don’t allow the tip to touch the pan’s bottom). If desired, insert a frying basket into the oil. Heat the oil to 350°F. Working with one batch at a time, take a large handful of potato slices from the water and dry them very well on paper towels. Fry each batch at 350°F. until softened, limp, and only beginning to turn golden, 4 to 6 minutes per batch. Remove the potatoes from the oil by lifting the handle on the frying basket, and carefully shake off any excess oil. Drain the potatoes on paper towels. (Alternatively, use a long-handled slotted utensil such as a wire spider to remove the potatoes from the oil.) Pile the potatoes into a large roasting pan lined with more paper towels and let the slices sit in the roasting pan at room temperature until you’re ready for the final frying before serving. Set the pot of oil aside.

When almost ready to serve, place a large, shallow baking sheet in the oven and preheat the oven to 325°F. Reheat the oil until it reaches 375°F. Make a deep slit in the garlic cloves without opening them up, and fry them in the hot oil until they’re golden and have flavored the oil, about 3 minutes. Remove them using your slotted utensil and set them aside. Fry the potatoes for the second time, in batches, until they’re golden and crisp, 3 to 5 minutes. Drain each batch on paper towels, then transfer them to the preheated baking sheet. Add the garlic to the hot oil, just long enough to reheat fully, then add to the platter of potatoes. Add the parsley to the hot oil, if using, and cook it for 10 seconds (the leaves will turn bright green). Use a triple-mesh wire sieve to retrieve the parsley and, after shaking it to remove any excess oil, add the parsley to the platter of potatoes. Use your hands to help the parsley and garlic filter down through the pile of potatoes. (At first, when you remove the parsley from the oil, it might seem limp, but it will become crisp very quickly. Don’t overcook the parsley or the leaves will taste acrid.)

Sprinkle the potatoes with salt and serve them hot with the spicy ketchup.