Place cucumber and chiles in a medium heatproof bowl. In a heatproof measuring cup, whisk ½ cup lime juice, 1 ½ teaspoons brown sugar, 1 ½ teaspoons salt, and ½ cup water. Microwave on high until simmering, about 1 minute 15 seconds. Stir to dissolve sugar, then pour over cucumber and chiles, pressing down to submerge. Place plastic wrap over surface of liquid to keep submerged. Let sit until ready to serve.
In a medium bowl, whisk 2 teaspoons fish sauce, ½ teaspoon salt, and remaining 2 tablespoons lime juice and 1 teaspoon brown sugar. Add cabbage, carrots, cilantro, and basil and toss to combine. Let sit until ready to serve.
In a large bowl, stir scallions, ginger, garlic, curry paste, soy sauce, and tamarind puree (if using) until combined. Add turkey and gently mix with your hands to combine. Form into 4 patties.
In a large skillet over medium heat, heat oil until shimmering. Season patties on both sides with a generous amount of salt. Cook patties, turning halfway through, until browned and cooked through, about 5 minutes per side.
Meanwhile, in a small bowl, combine mayonnaise, chili sauce, and chili crisp.
Spread cut sides of buns with chili mayo. Top bottom buns with a turkey burger, pickled vegetables, and cabbage slaw. Close with top buns.
