Pickle the vegetables: In a medium bowl, whisk the rice vinegar, warm water, 0.25 cup sugar, and kosher salt until dissolved. Add the carrot and daikon, press to submerge, and let stand at room temperature until lightly wilted and tangy, 20–30 minutes; drain well before assembling.
Make the spicy mayo: In a small bowl, stir together the mayonnaise, sriracha, and lime juice. Refrigerate until needed.
Form the patties: In a large bowl, gently mix the ground beef with the fish sauce, minced garlic, minced lemongrass, black pepper, and 1 tsp sugar just until evenly combined. Divide into 4 portions and form 4.5-inch patties with a shallow dimple in the center of each.
Cook the patties: Heat the neutral oil in a large skillet or on a grill over medium-high heat until shimmering. Cook the patties 3–4 minutes per side for medium (browned crust and an internal temperature of about 135°F), or to desired doneness; rest 5 minutes.
Toast the buns: Melt the unsalted butter in the skillet over medium heat, then place the split buns cut side down until golden, 1–2 minutes.
Assemble the burgers: Spread spicy mayo on the cut sides of the buns. Layer cucumber on the bottom buns, add a rested patty, top with a generous mound of drained pickled carrot and daikon, scatter jalapeño slices and cilantro sprigs, and close with the top buns. Serve immediately.