Preheat the oven to 350°F (170°C) and line a baking tray with parchment paper or grease it well.
Peel the potatoes and cut them into small dice.
Melt about one-third of the butter (2 tbsp) in a frying pan over low heat, add the diced potatoes, and cook, stirring occasionally, for about 15 minutes until just tender.
Melt the remaining 4 tablespoons of butter in a small pan or in the microwave and set aside.
Place the cheddar chunks in a food processor and pulse until crumbly.
Add the eggs, melted butter (4 tbsp), ricotta, chopped basil, chopped chives, salt, and pepper to the processor and pulse until the mixture is well combined.
Sift or sprinkle in the flour and baking powder, then pulse briefly until incorporated.
Spread the cooked potatoes in an even layer on the prepared baking tray.
Pour the egg-and-cheese mixture over the potatoes, spreading evenly, and bake for 20–25 minutes until browned on top and set through.
Allow the frittata to cool slightly, then cut into 20 squares and serve garnished with extra fresh basil if desired.