Lightly oil the grill rack and heat the grill to medium. Grill the pineapple for 5 minutes or until tender, turning it over once halfway through the grill time. Remove the pineapple to a cutting board. Dice the pineapple. Stir the pineapple and 2/3 cup picante sauce in a medium bowl.
Thoroughly mix the beef and the remaining picante sauce in a large bowl. Shape the beef mixture into 4 (1/2-inch-thick) burgers.
Grill the burgers for 10 minutes for medium or until desired doneness, turning the burgers over once halfway through the grill time. Top the burgers with the cheese.
Serve the burgers on the buns. Top with the pineapple-picante mixture and the avocado.