Step 1

Heat broiler. Broil shallots, garlic, and Anaheim and Thai chiles on a rimmed baking sheet, turning halfway through, until charred in most spots, 4–8 minutes. (Alternatively, you can char individually over a burner if you’ve got a gas stovetop.) Let cool.

Step 2

Peel shallots, garlic, and chile (it’s okay if some charred bits stay attached; they will add smoky flavor), then coarsely chop. Transfer to a mortar and pestle or a food processor and mash or pulse to a coarse paste. Mix in lime juice, cilantro, fish sauce, and sugar; season with salt. Taste and add more lime juice, fish sauce, and/or sugar if needed.

Step 3

Do ahead: Dip can be made 1 day ahead. Cover and chill.

Step 4

Combine chicken, shallot, lime leaves, curry paste, fish sauce, coconut milk, sugar, and 2 Tbsp. cilantro in a medium bowl; season with salt. Using your hands, mix until chicken is sticking to itself and mixture is smooth, about 3 minutes. Using wet hands, form about 2-Tbsp. scoops of chicken mixture into 1"-thick patties.

Step 5

Heat oil in a large skillet over medium-high. Working in batches, cook patties until golden brown, about 2 minutes per side. Transfer to paper towels to drain.

Step 6

Serve patties with chile dip, cabbage, daikon, cilantro, and rice.