Prepare a medium gas or charcoal grill fire.
In a medium bowl, gently mix the turkey, curry paste, and 1/4 tsp. each salt and pepper with your hands just until blended. Form the mixture into four 1/2-inch-thick patties.
Oil the grill grate. Grill the burgers, flipping once, until an instant-read thermometer inserted into the centers of the burgers registers 165°F, 12 to 15 minutes total.
While the burgers cook, finely grate 1 tsp. zest from the lime, then squeeze the lime to yield 1 Tbs. juice. In a small bowl, combine the zest and juice, mayonnaise, and 1/4 tsp. salt. In another small bowl, toss the cucumber with the cilantro, shallot, and vinegar.
Toast the buns cut side down on the grill until browned and heated through, 1 to 2 minutes.
Spread the tops and bottoms of the buns with the mayonnaise. Serve the burgers in the buns, topped with a slice of tomato and the cucumber mixture.