Trim fat from steak. Rinse meat, pat dry, rub with oil, and sprinkle liberally with salt and pepper.
Lay steak on an oiled cooking grate over a solid bed of very hot coals or highest gas heat (you can hold your hand 1 to 2 in. above grate for only 1 to 2 seconds); cover gas grill. Cook until firm when pressed on thin end but still quite pink inside (cut to check), 8 to 10 minutes; turn over halfway through. Transfer steak to a plate; let cool at least 30 minutes. About 45 minutes before slicing, put steak in the freezer (it will be easier to slice thinly).
On a board with a very sharp knife, cut steak across the grain, straight up and down and as thinly and evenly as possible, to make 30 to 34 slices; save narrow and uneven ends just to eat. If wet, blot with paper towels.
Measure accumulated meat juices and add enough milk to make 2 tbsp.; pour into a bowl. Crumble cheese into bowl and mash with a fork into a smooth paste.
Working with 1 steak strip at a time, lay strip on a flat side. Spread strip with about 3/4 tsp. cheese mixture, then lay 1 mint leaf and 1 mango piece at end of strip and roll to enclose. Let stand at room temperature at least 10 minutes before serving.
Note: Nutritional analysis is per roll.