To make the Pickled Daikon and Carrot, combine the daikon, carrot, vinegar, and sugar in a bowl, tossing well to coat. Set aside to marinate.
Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium-high. If a grill is unavailable, simply heat a grill pan over medium-high heat on your stovetop.
To make the Patties, combine the pork, garlic, diced chiles, ginger, fish sauce, soy sauce, five-spice powder, sugar, and lime zest and juice in a bowl, mixing well to incorporate. Divide the mixture into 6 portions and form into round patties.
Brush the grill rack with vegetable oil. Place the patties on the rack and grill for 6 to 7 minutes per side, until cooked through. Turn the patties 90 degrees halfway through cooking on each side, to attain grill marks.
Spread the cut side of each bun half with 1/2 tablespoon butter and grill until golden brown, 1 to 2 minutes.
Spread 1 tablespoon of the mayonnaise on the cut side of each bun bottom and 1 tablespoon of the pate on the cut side of each bun top. Place the cucumber and chile slices on the bun bottoms and top with burger patties. Drain the carrot and daikon pickles and pile a generous amount atop each patty. Add the cilantro sprigs and the pate-smeared bun tops. Enjoy!
Cook's note: These burgers can also be served on baguettes, if preferred, in which case, the patties should be formed into oblong shapes. Hamburger buns are a bit softer and easier to manage, which is why I've chosen them for this recipe. If available, buy a peppercorn pate, which adds a great hint of black pepper to the burger. For serving, burgers can be halved and skewered for a beautiful presentation and manageability. They go well with sweet potato french fries or taro chips!
A viewer, who may not be a professional cook, provided this recipe. The Food Network Kitchens chefs have not tested this recipe and therefore, we cannot make representation as to the results.