To make the Allioli, put the garlic, egg yolk, lemon juice, and honey in a bowl and whisk together. Slowly drizzle in the olive oil while whisking until you have a thick pale yellow emulsion.
Thoroughly dry the cut up potatoes using paper towels. Heat the duck fat in a heavy bottomed pan over medium heat until about 350 degrees F. I only had about half a cup of duck fat left, but if you have more fat, your potatoes will fry faster and you won't have to flip them. Put the potatoes in the oil and fry, stirring occasionally until they have just a hint of colour on the exterior. Transfer them to the paper towel lined rack and let them cool for a minute. This step removes the excess water near the surface of the potato, allowing them to crisp on the second fry.
Put the potatoes back in the hot oil. If you are shallow frying, leave the potatoes undisturbed until you can see the edges turning golden brown. Flip the potatoes over using tongs and brown the other side. When they're brown and crisp all over, transfer to a paper towel lined rack.
Pile the potatoes on a plate and sprinkle with pimentón, cayenne pepper, and sea salt. Drizzle with olive oil, then put a few dollops of allioli on top of your patatas bravas. Serve with some toothpicks.