Heat the oven to 350°F and grease a 2-quart baking dish with butter. Coat the bottom and sides so the peach syrup and pastry release cleanly after baking.

Melt the butter in a large saucepan over medium heat, then stir in the peaches, sugar, flour, lemon zest, lemon juice, cinnamon, ginger, and salt; cook for about 5 minutes, stirring often, before transferring the filling to the dish. The sugar should be fully dissolved and the fragrant juices should look thicker and glossy rather than thin and watery.

Keep the cinnamon rolls connected when opening the tube, set the glaze aside, and use a serrated knife to cut the dough crosswise into 1/2-inch rounds, making 10 pieces. Use a gentle sawing motion so the spirals stay round instead of being pressed flat.

Lay the dough rounds over the peaches, leaving tiny gaps where possible, and bake for 40 to 45 minutes. The fruit should bubble around the edges, and the roll tops should be deep golden with centers that feel springy rather than soft and gummy.

Rest the cobbler for 10 minutes, then spread the reserved glaze across the warm rolls. The brief cooling period lets the syrup settle so the icing melts into a soft layer instead of sliding into the fruit.