Bring onion and almost all of the water (save about 1 tablespoon for later) to a boil in a small nonstick skillet over medium-high heat. Cover and cook until the onion is slightly softened, about 2 minutes.

Uncover and continue cooking until the water evaporates, 1 to 2 minutes. Drizzle in oil and stir until coated. Continue cooking, stirring often, until the onion starts to brown, 1 to 2 minutes more.

Scramble the eggs and add to pan, reduce heat to medium-low and continue cooking, stirring constantly, until the egg is starting to set, about 20 seconds. Continue cooking, lifting the edges so the uncooked egg will flow underneath, until mostly set, about 30 seconds more.

Reduce heat to low. Sprinkle herbs, salt, and pepper over the frittata. Spoon cheese on top. Lift up an edge of the frittata and drizzle the remaining 1 tablespoon water under it.

Cover and cook until the egg is completely set and the cheese is hot, about 2 minutes. Slide the frittata out of the pan using the spatula and serve.