1. Stir gently to combine blackberries, sugar, and liqueur in a bowl. Let sit until berries soften, 15 minutes. Taste; add sugar if needed. Add the vanilla extract and mint. Refrigerate.
2. Adjust oven rack so grapefruit will be a few inches from the heat source. Preheat broiler.
3. Using a paring knife, cut around fruit, separating the “meat” from the pith. Cut along both sides of each dividing membrane so sections can be removed easily with a spoon. Line a baking sheet with aluminum foil and place grapefruit halves cut side up. Sprinkle with sugar (about 1 tsp per half).
4. Broil until the sugar caramelizes and turns golden brown, 2 to 5 minutes. (Watch carefully so it doesn’t burn.) Remove from oven and let sit a few minutes before serving. Serve topped with some of the blackberries. Drizzle blackberry juices over top. Garnish with mint.
Serves 6. Per serving: 160 calories, 40g carbs, 1g protein, no cholesterol or fat.