Preheat an oven to 220 degrees C. Line a baking tray with baking paper.
In a bowl, mix flour, baking powder, salt and cayenne pepper. Rub or cut the cold butter into the flour mixture until the mixture resembles fine breadcrumbs. Lightly stir in the grated cheddar cheese and the chives. Add as much milk as you need, a little at a time, until you get a soft, but not sticky dough.
Scrape the dough out onto a floured surface and gently knead a few times with your fingertips, just to flatten into an oblong shape, 10 cm wide, 15 cm long and about 4 cm high. Sprinkle 1 tablespoon of cheddar cheese on top of dough, and lightly press the cheese down into the top of the dough. Make 12 scones with a scone cutter or a glass dipped in flour.
Place the scones onto the baking paper-lined baking tray. For softer sided scones, move them about 1 cm apart; for crispier sides, move them farther apart. Bake for 10 to 12 minutes in the preheated oven, until the scones are golden and the cheese has melted and begun to brown. Serve hot with butter.
The dough will take between 3/4 and 1 cup (190 - 250ml) of milk, but you probably won't need a whole cup. You cut the scones into 8 big ones, or 12 small scones.