Preheat the oven to 400°F. Line two baking sheets with parchment paper.
Cut the potato in half and scoop the flesh of the potato into the bowl of a food processor with the buttermilk. Blend until smooth.
Place the flour, baking powder, sugar, and salt in the bowl of an electric mixer and mix on low speed for a few seconds to combine. Add the butter and mix on low speed for 3 to 4 minutes, until the butter is the size of peas. Gradually add the buttermilk mixture and blend until the dough comes together.
Divide the dough in half and knead each half on a lightly floured work surface to bring the dough together completely. Pat each piece of dough to a ¾-inch thickness. Using a 2-inch round biscuit cutter, cut out circles of dough and place on the prepared baking sheets, spacing them 2-inches apart. Brush the tops of the biscuits with the egg wash. Bake for about 20 minutes, until golden brown and firm to the touch.
While the biscuits cook, stir together the butter, goat cheese, and honey in a medium bowl to combine. Serve the warm biscuits with the goat cheese butter alongside.