½ cup water
¼ cup distilled white vinegar
¼ cup granulated sugar
1 tablespoon kosher salt
1 medium carrot, peeled and sliced into matchsticks or grated on the medium-hole side of a box grater
3-ounce piece daikon, peeled and sliced into matchsticks or grated on the medium-hole side of a box grater
¼ cup mayonnaise
1 jalapeño, finely chopped
1½ teaspoons white wine vinegar
¼ teaspoon kosher salt
2 tablespoons soy sauce
2 tablespoons sweet chile sauce (such as Mae Ploy)
1 tablespoon hoisin sauce
1 tablespoon rice vinegar
1 teaspoon freshly grated ginger
1 teaspoon fish sauce
1¼ pounds ground pork
4 scallions, trimmed and thinly sliced
1 bunch fresh cilantro, one-half finely chopped and one-half roughly chopped
¾ teaspoon kosher salt
1½ teaspoons freshly ground black pepper
2 teaspoons canola or grapeseed oil
4 crusty burger buns
4 slices pâté de campagne (about ¼ pound) or other rough-textured pork pâté (optional)
1. Make the pickled vegetables: In a small saucepan set over medium-high heat, bring the water, vinegar, sugar and salt to a boil. Turn off the heat and set aside to cool, about 15 minutes. In a medium bowl, add the carrot and daikon and pour the cooled liquid over the vegetables. Toss to combine and refrigerate.
2. Make the jalapeño mayonnaise: In a small bowl, stir together the mayonnaise, jalapeño, vinegar and salt. Cover with plastic wrap and refrigerate.
3. Make the glaze: In a medium saucepan set over medium-high heat, bring the soy sauce, chile sauce, hoisin sauce, rice vinegar, ginger and fish sauce to a simmer. Reduce the heat to medium-low and simmer until the sauce is slightly thick and coats the back of a spoon, about 10 minutes. Turn off the heat and set aside.
4. In a medium bowl, combine the pork, scallions, finely chopped cilantro, salt and pepper. Divide the mixture into 4 equal parts and shape into 1-inch thick patties. Heat a grill pan for 2 minutes over medium-high heat, add the oil to the pan and then add the patties. Grill until browned, 3 to 4 minutes, then use a spatula to flip the burgers. Liberally brush the grilled side with some of the glaze and continue to cook until the other side of the burger is browned and the center of the burger resists light pressure, 3 to 4 minutes more. Turn the burgers and brush with more glaze. Cook for 1 minute longer, then turn off the heat.
5. Place the burger buns on a platter. Spread 1 tablespoon of the mayonnaise on each side of the bun. On the bottom side of the bun place a burger and top with the pâté, a pile of pickled vegetables, the roughly chopped cilantro and the top half of the bun. Serve.