In a 9-inch ovenproof nonstick skillet, bring 1/4 cup of water to a boil over moderately high heat. Add the spinach and cook, stirring, until wilted, about 1 minute. Using tongs, transfer the spinach to a colander. Repeat with the arugula. Gently squeeze the greens dry and coarsely chop them. Wipe out the skillet.
Preheat the oven to 400. In a bowl, whisk the eggs, egg whites, cheese and a large pinch each of salt and black pepper.
Heat 3 teaspoons of the oil in the skillet. Add the shallot and sliced pepper and cook over moderate heat, stirring, until the pepper is just tender, 4 minutes. Add the thyme and the chopped greens and cook, stirring, until hot. Season with salt and black pepper.
Add the remaining 1 teaspoon of oil to the vegetables, then pour in the egg mixture. Tilt the skillet to evenly distribute the eggs and cook over moderately low heat until set around the edges, 3 minutes. Bake the frittata on the top shelf of the oven for 4 minutes, until just set. Invert the frittata onto a plate and let cool slightly. Cut into wedges and serve.