1. Heat the oil in a large skillet over medium-high heat.
2. Add the garlic and cook for 30 seconds.
3. Add the spinach and cook, tossing, until wilted, 2 to 3 minutes.
4. Season with 1/4 teaspoon salt and 1/4 teaspoon pepper.
5. In a small bowl, combine the spinach with the salmon.
6. Wipe out the skillet and melt the butter over medium-high heat.
7. Season the eggs with the remaining salt and pepper.
8. Pour the eggs into the skillet and swirl the pan so they spread out.
9. Using a heat-resistant rubber spatula, mix the center of the eggs quickly once or twice, then lift the eggs around the edges occasionally to let any uncooked portion flow underneath. Cook until the bottom is firm.
10. Sprinkle the cheese evenly over the eggs, then arrange the spinach-salmon mixture over the cheese; cook for 20 seconds more.
11. Slide the omelet onto a plate, carefully folding the egg over the filling as you do. Serve immediately.