1. Position a rack in the middle of the oven and preheat to 375 degrees F. Butter and flour an 8-inch cake pan. Line the bottom of the pan with a round of parchment.
2. Sift together the flour, baking powder, baking soda and salt into a mixing bowl. In another large bowl, beat the butter and granulated sugar with an electric hand-held mixer at medium-high speed until pale and fluffy, about 2 minutes. Add the egg and vanilla and beat well. At low speed, mix the flour mixture into the eggs in three batches, alternating with the buttermilk, beginning and ending with the flour, until just combined. Scoop the batter into the cake pan. Smooth out the top and scatter the cherries and hazelnuts evenly over the batter. Sprinkle with the turbinado sugar.
3. Bake the cake until golden brown and a tester inserted into the center comes out clean, 30 to 35 minutes. Cool in the pan 10 minutes, and then turn out onto a rack, remove the parchment, invert and cool completely. Serve.