Rub the mesquite grill seasoning evenly over all sides of the chicken breasts. Place on a plate, cover with plastic wrap and refrigerate for at least 30 minutes.
Preheat grill to high, and cook chicken breasts on greased grill grates, flipping occasionally, until cooked through, about 10 minutes total. Place two bacon slices onto each chicken breast, and top with a slice of cheese. Close grill lid and let cheese melt, about a minute. Remove from grill.
Whisk together the mayo, mustard, and Sriracha. Spread evenly on the top of a bun, and the stack chicken breast with tomato and lettuce.