Preheat oven to 425 degrees and place a 10-inch cast-iron pan into the oven while it's preheating.
Pour flour, baking powder, and salt into a medium-sized bowl. Cut 4 T of the butter into small pieces and add it to the bowl, with the goat cheese and the yogurt. Stir until the mix is moistened, adding an extra tablespoon of yogurt if needed.
Remove the hot skillet from the oven and place a tablespoon of butter into it. When the butter has melted, divide the batter into 12 biscuits, each about the size of a golf ball, and then nestle them into the pan-they'll be snuggled in there pretty cozily, and that's OK, (about 9 around the edge and 3 in the middle.
Brush the tops of the biscuits with one tablespoon of melted butter. Bake for 14 to 16 minutes until browned on the top and bottom. Remove from the oven and sprinkle with the grated parmesan cheese.