1. Trim ends of zucchini, then coarsely grate on large holes of a box grater. Toss zucchini with 3/4 tsp salt in a large bowl and let stand 30 minutes.
2. Transfer zucchini to a colander, then firmly squeeze handfuls to remove excess liquid.
3. Heat olive oil in a 10-inch heavy skillet over moderately high heat until hot but not smoking and sauté zucchini, stirring until golden, 6 to 7 minutes. Remove skillet from heat and stir in marjoram, then let mixture cool to warm, about 15 minutes.
4. Lightly beat eggs with zucchini and pepper, in a large bowl, using a fork.
5. Heat butter in a 7- to 8-inch nonstick skillet over moderately high heat until foam subsides and butter has a nutty fragrance. Add egg mixture, distributing zucchini evenly with a heatproof rubber spatula, and cook, lifting up egg around edges occasionally to let any uncooked egg flow underneath, until egg mixture is set around edge, about 1 minute.
6. Reduce heat to moderately low and cook omelet until softly set but top is still moist, about 3 minutes.
7. Shake skillet to loosen omelet from pan, then slide omelet onto a large plate.
8. Wearing oven mitts, invert skillet over omelet, then holding skillet and plate together invert omelet, browned side up, into skillet. Cook omelet until underside is set, about 1 minute, then slide omelet onto a serving plate.