Place 1 medium red onion, thinly sliced, and one 3" piece ginger, peeled, cut into thin matchsticks, in a medium bowl of ice water and let soak while you prepare the chicken and rice.

Using your hands, toss 2 cups cooled cooked jasmine rice and half of one 4-oz. can or jar Thai red curry paste (no need to be exact) in a medium bowl to coat. Set aside.

Stir 3 Tbsp. fresh lime juice, 1 Tbsp. plus 1½ tsp. fish sauce, and 1 Tbsp. plus 1½ tsp. sugar in a large bowl until sugar is dissolved.

Heat 2 Tbsp. vegetable oil in a large nonstick skillet over medium-high. Add 1 lb. ground chicken and spread into a thin, even layer; season with kosher salt. Cook, undisturbed, until golden brown underneath, about 5 minutes. Add remaining 2 oz. Thai red curry paste and cook, stirring and breaking up into bite-size pieces, until chicken is cooked through, about 2 minutes. Transfer to bowl with dressing.

Heat remaining 2 Tbsp. vegetable oil in same skillet. Add reserved rice mixture and press into an even layer. Cook, undisturbed, until golden and crisp underneath, 6–8 minutes. Continue to cook, breaking up rice into large pieces, 1–2 minutes. Remove from heat.

Drain red onion and ginger; add to bowl with chicken, along with 1 small serrano chile, thinly sliced, 1 cup salted dry-roasted peanuts, 1 cup (packed) cilantro leaves, ½ cup (packed) mint leaves, and rice. Toss well to combine.

Transfer salad to a platter. Serve with lettuce leaves for wrapping and lime wedges for squeezing over.