Heat a 10-inch cast-iron skillet over medium. Add the blackberries, 1/2 cup granulated sugar, the brown sugar and the liqueur. Cook, stirring constantly and lightly crushing the berries, reducing the heat as needed to prevent burning, until a loose jam forms, about 25 minutes. Remove from the heat; stir in 1 tbsp. butter. Let cool.
Preheat the oven to 325 degrees . Sift the flour, cocoa powder, baking powder and 1/4 tsp. salt into a medium bowl. In a medium metal or glass bowl set over a saucepan of gently simmering water, heat the chocolate and the remaining 6 tbsp. butter, whisking frequently, until smooth, 3 to 5 minutes. Whisk in the very hot water; remove the bowl from the saucepan. Whisk in 1 cup granulated sugar (the mixture will be grainy). Whisk in the eggs, one at a time, beating to blend between additions, then whisk in the vanilla.
Whisk half the dry ingredients into the chocolate mixture. Whisk in the buttermilk, then the remaining dry ingredients.
In a medium bowl, gently toss the raspberries, lemon juice and remaining 1/4 cup granulated sugar. Scatter over the blackberry jam in the skillet. Spread the batter on top in an even layer.
Bake the cake, rotating the skillet halfway through baking, until the juices bubble at the edges and a tester inserted into the center comes out clean, about 40 minutes.
Let the cake cool in the skillet 15 minutes, then run a butter knife around the edge to loosen. Place a serving plate on top of the skillet; carefully invert the cake onto the plate. Slice into wedges and serve with vanilla ice cream, creme fraiche or fresh whipped cream.