In a food processor, combine all the burger ingredients except ground turkey.
Process until finely mixed into a paste, adding more water as needed.
In a bowl, combine ground turkey with the paste and stir with a wooden spoon, spatula, or your hands.
Shape into ¾-inch-thick patties.
Arrange patties on a baking sheet and refrigerate for at least 1 hour.
While the patties are in the fridge, make the spicy mayo: In a bowl, mix mayonnaise, green sriracha sauce, and lime juice. Taste and adjust for green sriracha and lime juice.
Make the slaw: Mix carrot slaw with 2 tablespoons of spicy mayo, or more as needed.
Cook the patties in a skillet set over medium heat, turning once, for about 10–12 minutes, until they reach an internal temperature of 180 F.
Toast the buns in an oven set to the grill or broil setting for 5 minutes.
Spread a generous amount of spicy green sriracha mayo on both sides of each bun.
Place lettuce leaves on the bottom of each bun.
Top lettuce with cooked turkey burger.
Layer each burger with a generous amount of cabbage slaw and 4–5 Thai basil leaves.
Place the top half of the bun on top, pressing down slightly to hold the toppings in place. Repeat with remaining burgers.