To start with fresh tart cherries:
1. Wash the cherries, remove and discard their stems, remove and reserve their pits.
2. Put cherries in a bowl and mix with sugar 8:1 (by volume). Let the sugar draw out cherry juices for about 3 hours at room temperature or overnight in the refrigerator. Strain the juice and reserve it for later use.
3. Break reserved pits. To extract the flavor, place broken pits in a sauté pan, add the strained cherry juice, and simmer until reduced by half and slightly thickened. Cool it down to room temperature and strain to remove pit fragments. Allow the syrup to cool before using it in Step 5.
4. Gently squeeze extra moisture out of drained sour cherries. You can sprinkle them with some corn starch. Added starch will absorb and thicken leftover cherry juices inside the cooked dumplings.
5. While varenyki are boiling or steaming, prepare the sauce. Mix some corn starch with the tart cherry syrup (1/2 tsp starch per 1 cup of liquid) and slowly bring to simmer. Add more pitted tart cherries or other berries to the sauce at this point. Cook for 2 minutes to let the sauce thicken. Taste it to make sure the flavor of raw starch is gone. The sauce and varenyki should be ready for serving at the same time.