1. Brush 1 tablespoon oil on each cut side of roll tops. Spread out sides of roll bottoms with jam, top with cheese, then ham and basil; cover with tops.
2. Toast in a panini machine (following manufacturer's directions) or in a lightly oiled grill pan over moderate heat (pressing down on sandwich with a heavy skillet or sturdy metal spatula) turning once, until bread is golden brown and cheese begins to melt, about 4 minutes.
3. Cut sandwiches in half and serve immediately. Makes 8 sandwiches.