Make the blackberry sauce: In a small saucepan, combine blackberries, 2–3 tablespoons honey (or sugar), a pinch of salt, and half the lime zest.
Cook over medium heat for 5–7 minutes, stirring and gently mashing, until the berries break down and release their juices.
Thicken slightly: Let the mixture simmer another 2–3 minutes until syrupy. It should coat the back of a spoon but still flow. Remove from heat and stir in 1 tablespoon lime juice.
Taste and adjust sweetness.
Cool the sauce: Set aside to cool to room temperature. For faster cooling, spread it on a plate and refrigerate for 10 minutes. The sauce should be cool before swirling to keep the yogurt from melting.
Flavor the yogurt: In a bowl, whisk Greek yogurt with 1–2 tablespoons honey (or to taste), vanilla extract, remaining lime zest, and a small squeeze of lime juice.
The yogurt should taste bright and lightly sweet.
Assemble the swirl: Spoon half the yogurt into a shallow bowl or glasses. Add a few spoonfuls of blackberry sauce. Layer with the remaining yogurt, then more sauce.
Create the pattern: Use a butter knife or skewer to gently drag through the layers in figure-eights. Don’t overmix or you’ll lose the defined swirl.
Finish and serve: Top with a few whole blackberries, a sprinkle of lime zest, and optional mint.
Serve right away, or chill for up to 30 minutes for a firmer, dessert-like texture.