Heat oven to 350°F. Butter a 9×13-inch baking dish or 12-inch Dutch oven.
Melt butter in a saucepan. Stir in both sugars, cinnamon, Mountain Dew, and both whiskeys. Simmer gently for 5 minutes, stirring occasionally.
Cut apples into 16 thick wedges. Separate dough triangles and sprinkle with cinnamon and sugar.
Roll one apple wedge into each triangle, starting at the wide end. Arrange in the dish.
Add sauce halfway up the dumplings, reserving 1–1½ cups. Bake about 40 minutes, until browned.
Mix cornstarch with cold water. Bring reserved sauce to a gentle boil, stir in slurry, and thicken. Spoon over dumplings.
Optional: Replace whiskey with additional Mountain Dew.