With the motor running on a food processor, drop in the garlic and ginger and process until finely chopped. Add pancetta, onion, chile flakes, soybean paste, chile paste, sugar and pepper and process until fairly smooth. Set the pancetta mixture aside.
Crumble the beef into a large bowl. Add the seltzer, sesame seeds, pancetta mixture and some salt and pepper and gently mix together with your hands, being careful not to overwork the mixture. Form the mixture into 4 patties, each about 1 inch thick and 4 inches wide. Create a depression in the center of one side of each patty, as burgers tend to rise in the middle during cooking and this will help the burgers come out flat. If not cooking immediately, cover the patties and refrigerate.
Heat the oil in a large skillet over medium-high heat. Put the patties in the skillet depression-side up and cook, flipping halfway through, until browned and cooked through, about 7 minutes total.
Meanwhile, heat a two-burner griddle/grill pan or skillet over medium-high heat. Spread both sides of the buns with the butter and cook cut-side down until lightly toasted, about 1 minute. (If working in batches, toast the bottom buns first.) Transfer to individual plates.
Put a patty on each bottom bun and top with lettuce and then the Cucumber Kimchi. Smear some Korean Ketchup and Doenjang Mayonnaise on the top buns and place on the burgers. Secure with bamboo skewers or long toothpicks and serve immediately.
Yield:about 4 cups
Cut each cucumber crosswise into 2-inch slices with a small knife. Stand the slices on their cut sides and cut each one two-thirds of the way down into quarters, keeping them attached at the bottom. Sprinkle the cucumbers with the salt, spreading the cucumbers open to get the salt deep inside the cuts. Arrange the cucumbers with their cross-cut sides up in a single layer in a glass or other nonreactive container that is at least 2 inches tall and that has a tight-fitting lid. Cover and let soften at room temperature for 30 minutes to 1 hour.
Meanwhile, combine the scallions, onion, chile flakes, shrimp, garlic, ginger and 2 tablespoons water in a food processor. Pulse until a coarse spice paste forms, then stir in the chives.
Rinse the salted cucumbers well under cold water, making sure to rid the crevices of all the salt. Shake dry and then press the spice paste all over and into the crevices of each piece. Return the cucumbers to the (rinsed) container, cross-cut sides up, packing them somewhat tightly and pressing in any remaining spice paste and liquid. Cover and let the cucumbers ferment at room temperature for about 24 hours. Refrigerate until ready to serve.
Stir together the ketchup and chile paste in a small bowl. Cover and store in the refrigerator if not using immediately.
Whisk together the mayonnaise and soybean paste in a small bowl until smooth. Cover and store in the refrigerator if not using immediately.