Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
Toss 1 tablespoon of the sugar and the cornstarch with the lychees and strawberries in a large bowl until combined. Mix together the remaining 2 tablespoons sugar and the cinnamon in a small bowl.
Lay a spring roll wrapper on a work surface with a corner pointing toward you and brush the surface lightly with melted butter. Put 1 heaping tablespoon of the lychee mixture into the center of the wrapper. Bring the two side corners together so they meet over the filling. Bring the bottom corner up over the filling and roll the wrapper tightly around the fruit. Set the spring roll on the baking sheet with the exposed corner down; continue rolling with the remaining fruit and wrappers.
Brush butter all over the surface of the rolls and sprinkle each with a little of the cinnamon sugar. Bake until light golden brown and crisp, about 10 minutes. Do not overbake - the fruit will begin to break down as it cooks and can ooze out of the wrappers.
Let cool for at least 10 minutes before serving with a scoop of ice cream alongside.