Preheat oven to 350 degrees. Core and cut peppers into 1/2 inch strips. Place them on a sheet pan and roast oil-free (better for the flatbread) until they soften and slightly char, approximately 30 minutes.
As the peppers roast, start the flatbread dough: Place the flour in a bowl with 1 of the 2 tablespoons of oil, 1/4 teaspoon of salt and the room temperature water. Stir with a spoon to start, but you'll need to finish with your hands.
Knead the dough for 30 seconds on a lightly floured surface, then roll into a ball and place back into the bowl. Cover with a slightly damp towel to prevent crusting and rest for 15 minutes.
After those 15 minutes, remove the peppers from the oven and let cool.
Take the dough ball and cut into 2 pieces. Fold a small portion of the roasted veggies into each dough ball and re-roll.
Using a rolling pin, flatten each ball on a lightly floured surface to get to less than 1/4 inch thickness.
In a dry, nonstick frying pan, toast each flatbread on medium heat until golden, then place on separate plates.
Add the remaining 1 tablespoon oil to a frying pan and swirl to cover, and reduce heat to medium-low. Crack both eggs into the pan, sprinkle with the remaining salt and pepper, lightly cover and cook to desired degree.
While the eggs cook, distribute the spinach and remaining roasted peppers between the two flatbreads. Top each with a fried egg.