Heat the oven to 350°. Butter a 9-by-2-inch round cake pan. Line the pan with a piece of parchment paper cut to fit the bottom and butter and flour the paper.
Place the sugar and almond paste in a food processor and pulse until it's blended, about 40 seconds. Add the butter, zest, and vanilla extract and pulse until smooth, about 40 seconds. With the processor running, add the eggs one at a time. Remove the lid and add the flour, baking powder, and salt, then pulse a few more times, just until blended, about 15 seconds. Scrape the batter into the prepared pan and smooth the top.
Bake the cake until golden and a toothpick inserted into the center comes out clean, 45 to 50 minutes. Meanwhile, place the berries, honey, and vinegar in a medium bowl and stir to combine. Refrigerate for at least one hour or until the berries are juicy.
Cool the pan on a wire rack for 30 minutes, then loosen the cake's sides with a knife, invert it onto a serving plate, and peel off the parchment. Serve the cake dusted with confectioners' sugar and topped with the berries.