Trim the pineapples and quarter them lengthwise. Cut away the cores and rinds of each quarter and cut the pineapple into 1-1/2-inch pieces. Working in batches, pulse the pineapple in a food processor until crushed or about the size of grains of rice (it's fine if a few pieces are larger or smaller).

In a large nonreactive saucepan, bring 3/4 cup of the sugar and 2 cups of water to a boil over high heat. Add the pineapple and let the mixture return to a hard boil. With a spoon, skim away and discard any foam that rises to the surface. Boil for 1 minute and then remove the pan from the heat. Add the mint sprigs and let the mixture cool to room temperature. Remove and discard the mint. Cover and refrigerate until well chilled, at least 4 hours and up to 2 days.

One hour before serving, in a small bowl, mix the strawberries with the remaining 2 tablespoons sugar and refrigerate.

To serve, drain the liquid from the pineapple in a sieve set over a bowl. Divide the pineapple into clear glasses, saving the syrup for another use. Top with the strawberries and lychees and garnish with a sprig of fresh mint.