Step 1Remove about three-fourths of the zest from one of the oranges with a large-grade zester or peel strips of zest with a vegetable peeler and cut into a fine julienne. Measure one-half cup sugar and the water into a saucepan and bring to a boil to dissolve the sugar. Add the tiny strips of zest, reduce the heat, and simmer for about 30 minutes, until the peel is tender and the syrup has reduced and thickened slightly. Let cool.
Step 2Rinse the strawberries and dry them gently. Hull and slice them about one-fourth-inch thick into a bowl. Cut away all the peel from the oranges: Slice off the tops and bottoms and cut away all the rind. Remove skinless sections of the oranges by sliding a sharp paring knife alongside the membranes and prying out the segments into the bowl with the strawberries. Squeeze the juice from the remaining oranges, sprinkle with 2 tablespoons sugar, and mix gently.
Step 3To serve, spoon the compote into serving glasses or dishes, sprinkle some curls of the candied peel on top, and drizzle a teaspoon of the syrup from the peel over the compote.