Preheat oven to 400 degrees F. Line a baking sheet with parchment paper.
Make garlic butter. Melt 1 tablespoon butter in a small saute pan over medium heat. Add garlic and saute until aromatic and lightly browned, about 1-2 minutes. Remove from heat and transfer garlic butter to a bowl. Chill in the freezer until cold, about 10 minutes.
Combine chilled garlic butter with flour, sugar, baking powder, salt, sharp cheddar, parmesan, chives, and butter in a food processor. Add butter and pulse mixture until butter is cut into small chunks and incorporated into flour.
Add buttermilk and pulse about 5 times or until dough is just starting to come together. Transfer dough to a clean work counter and knead a couple of times until dough comes together.
Pat down dough into ½-inch thick square. Cut into 4 equal squares and stack the portions on top of one another. Press down on dough and pat down until ½-inch thick. Cut into 4 equal squares and repeat. Pat down dough until it's 1-inch thick. Wrap in plastic wrap and chill in the fridge for at least 30 minutes or up to 12 hours.
Remove dough from oven and cut into 6 equal portions. Lay biscuits on prepared baking sheet, spacing them 1-inch apart. Brush tops with melted butter and sprinkle grated parmesan on top. Bake biscuits for 20-25 minutes or until golden brown. Remove from oven and let cool slightly before enjoying.