Preheat oven to 400°F. Line a baking sheet with parchment paper. Unroll dough onto baking sheet. Flatten so that it is about 12 to 12½ inches in diameter. Combine 1 tablespoon of sugar with the cornstarch, stirring with a fork or whisk to get rid of any clumps. Sprinkle this mixture over dough, leaving a 2-inch border around the edges.
Combine the pitted cherries with the remaining 3 tablespoons of sugar, lime juice, and lime zest. Stir until combined and cherries are coated with sugar and lime.
Carefully arrange cherry mixture (use a spoon!) over dough, leaving a 2-inch border around the edge. Reserve about a cup of the mixture and the juice.
Fold the dough border over the cherries, folding and pressing to seal as you go. There will be an opening in the middle. Carefully add remaining cherries and juice to the middle.
Brush the edges of the dough with cream. Sprinkle turbinado sugar over the cherries and the edges of the dough.
Bake for 25 minutes or until crust is browned. Let cool on pan for at least 20 minutes prior to cutting and serving.