Make berry jam: Combine berries and honey in a small saucepan over medium heat and cook, stirring occasionally, until berries have thawed and honey has melted, about 2 minutes. Reduce heat to medium-low and simmer, stirring occasionally, until syrupy, about 13 minutes. Stir in vinegar. Remove from heat and let cool to room temperature. Chill until cold, at least 1 hour. Store in the refrigerator in an airtight container until ready to use.
Make frozen yogurt: Place an 8-by-4-inch freezer-safe container in freezer for at least 30 minutes. Whisk together yogurt, half-and-half, honey, and vanilla in a bowl until completely smooth, about 1 minute. (Mixture should be cold before whisking. If it has warmed at all, chill until cold before whisking.)
Pour yogurt mixture into freezer bowl of a 1- to 1 1/2-quart electric ice cream maker and freeze according to manufacturer's instructions until it forms the consistency of a thick milkshake, 17 to 19 minutes. Spread half of yogurt mixture in frozen container.
Dollop half of berry jam on top and swirl into yogurt mixture using a cold knife. Repeat with remaining yogurt mixture and berry jam. Gently press a piece of parchment directly onto yogurt mixture; wrap container tightly with plastic wrap. Freeze until easily scooped, at least 4 hours.
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