Special equipment: 6 ramekins
Preheat the oven to 350 degrees F.
In the food processor (or, alternatively, in a bowl), blend the flour, sugar and salt. Add the butter and pulse to blend (or mix in with your fingers) until the mixture is crumbly. Add the yolk, blend it in. If the dough does not hold together when pressed between 2 fingers, add the ice water, 1 tablespoon at a time, until it does. Turn the dough onto a lightly floured flat surface. Knead the dough until smooth, flatten it and wrap in plastic. Refrigerate.
Put the mangoes on a flat surface and, one by one, hold the mangoes upright and cut one slice off each "side". It's like quartering apples only you are avoiding the large pit in the center of each mango. Make each slice as deep as you can to extract the maximum flesh from each mango. Slice the smaller piece off the top and bottom of each fruit as well.
In a skillet over low heat, melt the sugar until it becomes golden brown. Add the butter (it will bubble up). Remove the skillet from the heat and pour equal amounts of the caramelized sugar mixture into each ramekin. Divide the mango pieces among the ramekins and press the fruit into the caramel sauce.
Roll the dough on a floured surface (or, alternatively, between 2 pieces of waxed paper) and cut out circles that fit the tops of the ramekins. Tuck the dough edges inside the ramekins so it covers the fruit. It is as if you are tucking the fruit into a pastry bed.
Place the ramekins on a sheet pan and into the center of the oven and bake until the pastry browns, 15 to 18 minutes. Remove the ramekins from the oven and allow it to cool for 10 minutes.
Unmolding: Don't be afraid of failure. A tarte tatin is nice when imperfect! Place a plate over the pastry topping and flip the tart on to the plate. You can do it! Pour the reserved caramel back over the tart. If any fruit remains in the bottom of the skillet, simply dislodge it and insert it any fruit gaps before serving. Serve warm with ice cream or a dollop of sour cream.