Snip enough basil to equal 1/3 cup. In large bowl combine half the snipped basil and ground ham. Shape in four 3/4-inch-thick patties.
Grill patties on rack of charcoal or gas grill, covered, directly over medium heat for 14 to 18 minutes, turning once, until patties register 160 degrees F with instant-read thermometer. Add pineapple to grill during last 6 minutes of grilling time for patties, turning once.
Meanwhile, in bowl combine mayonnaise, mustard, honey, and remaining snipped basil. Spread on cut side of roll bottoms. Layer patties, remaining basil leaves, pineapple, and roll tops.