For the tomato-basil pesto: Combine the basil, Parmesan, pine nuts, garlic, and salt and pepper in a food processor and process until smooth. With the machine running, drizzle in the olive oil. Transfer to a bowl and fold in the tomatoes. Set aside.
For the frittata: Put the potatoes in a saucepan, cover with cold water and add a few tablespoons of salt. Bring to a boil and cook until a skewer inserted into the center meets with no resistance, about 25 minutes. Drain and let cool slightly. Cut the potatoes into 1-inch cubes.
Preheat the broiler. Heat the oil in a large nonstick skillet over medium heat until it begins to shimmer. Add the potato cubes, season with salt and pepper, stir in the thyme and chile flakes, and cook until golden brown all over.
Whisk together the eggs until light and fluffy, then pour over the potatoes. Scatter the smoked mozzarella on top and season with salt and pepper; stir and briefly fold everything together. Cook until the egg mixture has set on the bottom and is beginning to set up on top. Transfer to the broiler and cook until lightly browned and fluffy, 5 to 6 minutes. Remove the frittata from the oven and carefully slide it onto a cutting board. Cut into wedges and serve hot or at room temperature with some of the pesto and a sprinkling of grated Parmesan.