In a medium bowl, mix 1 tbsp. granulated sugar, the cream and pepper. Cover; chill overnight.
Preheat the oven to 350 degrees . Grease a 9-inch springform pan with butter; line with a parchment-paper round. In a small saucepan, stir 1/2 stick butter with the brown sugar over medium heat until the butter melts, 2 to 3 minutes; pour into the cake pan. Arrange about half the tomatoes, cut side down, in concentric circles on top of the sugar mixture. Top with another layer of tomatoes, cut side down, filling in any spaces left from the first layer.
In a medium bowl, whisk the flour, baking powder, ginger and salt. In a liquid measuring cup, combine the milk and vanilla. In a large bowl, using an electric mixer, beat the remaining stick of butter and 1 cup granulated sugar on medium-high until pale, 5 to 6 minutes. Add the eggs, one at a time, beating to blend between additions. Beat in the dry ingredients in three additions, alternating with the milk in two additions, starting and ending with the dry ingredients. Pour into the pan.
Place the pan on a rimmed baking sheet lined with foil and bake until a toothpick inserted into the center comes out clean, about 1 hour 10 minutes. Let cool in the pan on a wire rack, 30 minutes. Invert the pan onto a platter and remove the pan sides. Drizzle the cake with any syrup that may have dripped onto the baking sheet. Let the cake cool completely.
Using an electric mixer, beat the black-pepper cream until soft peaks form. Cut the cake and divide among plates. Serve with a spoonful of lemon curd and a dollop of whipped cream.