1) Combine the butter, onion, and sugar in a pan set over medium-low heat. Cook the onions, covered, for about 30 minutes, stirring every 5 minutes; they should be nicely caramelized. Transfer the onions to a bowl, cool to room temperature, then refrigerate until well chilled, at least 3 hours.
2) Preheat the oven to 425°F. Grease or line a baking sheet with parchment.
3) Combine the flour, baking powder, and salt. Work the butter into the flour until the mixture is unevenly crumbly.
4) Toss in the chives and caramelized onions.
5) Cut in the starter until the dough becomes cohesive. Turn the dough out onto a lightly floured surface and use a bench knife or bowl scraper to fold it over on itself five or six times, until it comes together.
6) Pat the dough into a 1"-thick disk.
7) Use a sharp 2 1/2" biscuit cutter to cut rounds. Pat the scraps together, and cut additional biscuits.
8) Bake the biscuits for 15 to 18 minutes, until they're just turning golden.
9) Remove the biscuits from the oven, and serve warm with a little bit of butter.
Yield: eight 2 1/2" biscuits.