The ataulfo mangoes here are definitely the best choice. And I think I'm in the minority, but I do like @MokaPotdoes, I peel first (though not in a circular pattern) and then cut the cheeks off, then the remaining bits. Or I just chew the remaining bits off. 😃
That hedgehog cut with peel attached never seems to work very well for me, I don't know why.
I start by cutting a bit off one end so that i have a flat spot to balance the fruit after peeling and before cutting.