Pound chicken breasts until they are about 1/4" thick all around. Place in a large Ziploc bag. Combine Teriyaki sauce and brown sugar until sugar dissolves. Add 1/2 of the mixture to the bag with the chicken and seal tightly. Place bag in refrigerator for at least 4-5 hours. Separate remaining Teriyaki mixture into 2 separate bowls (to prevent cross contamination use 2 separate brushes too)-- one for the pineapple, one for the chicken. Open pineapple can and save a little of the juice in a bowl. Place chicken and pineapple on the grill (medium heat).
Brush with Teriyaki sauce every few minutes until chicken and pineapple are done-- try to only flip the chicken and pineapple once. Either on the grill or in a frying pan, cook ham with a little of the pineapple juice until it becomes nice and hot. Slice open hamburger buns and place a piece of cooked chicken on each bottom bun. Top with cheese, ham, grilled pineapple, and bun top. Serve immediately with remaining grilled pineapple.