In a large bowl, mix salt, paprika, cayenne, and a few grinds of black pepper. Rub chicken in spice mix until coated.
Cover bowl with plastic wrap. Refrigerate at least 30 minutes and up to overnight.
In a medium pot over medium-high heat, bring soda to a boil. Boil, stirring occasionally and watching closely, until thickened and reduced to about ¾ cup, 18-25 minutes. Whisk in soy sauce, Worcestershire, ketchup, brown sugar, lime zest and juice, and salt.
Prepare a grill for medium-high heat; preheat 5 minutes (or heat a grill pan over medium-high heat). Brush grates or pan with oil. Using tongs, arrange chicken on grill. Brush exposed side of chicken with glaze.
Grill chicken for 4 minutes. Flip, brush with glaze, and continue to grill for 4 minutes. Flip and brush one more time. Continue to grill until an instant-read thermometer inserted into thickest part of chicken registers 165°, about 4 minutes more.
Transfer to a platter and let cool 5 minutes. Serve with lime wedges alongside.
Make Ahead: Glaze can be made 5 days ahead. Transfer to an airtight container and refrigerate.
