That makes mangoes a challenge to cut-up when no one has ever told you the best way to do it. Allow me to be that influential person!
Hold the mango like a football that's about to be kicked.
Choose a side and begin to slice downward about an inch away from the center. If you hit the pit, move the knife over and start again, but get as close to the pit as possible and still be able to cut all the way through the mango.
With your knew mango piece, run the knife in a crosshatch pattern to score it. Try not to cut through the skin.
While successful, those chunks were big and had to be cut further for my Southwestern Bean Salad recipe.
Here's a picture of my second try, making the scoring marks closer:
Ah, that's better!
Then just pick it up and run the knife at the bottom of the scoring marks, just at the line of the skin. the hunks just fall right off.
Continue with the rest of the mango.