Heat 1 teaspoon oil in an 8-inch nonstick skillet over medium-high. Add mushrooms; sauté until golden, about 3 minutes. Stir in kale and half of salt; stir about 30 seconds or until kale is just wilted. Place the mushroom mixture on a plate or in a small bowl. Wipe pan clean.
Break eggs into a small bowl. Use a fork to beat them like Will Ferrell's cowbell until completely blended and slightly frothy. Stir in remaining salt and pepper.
Heat remaining 1 teaspoon oil in the same pan, over medium now. Add eggs; stir them around the pan with a heatproof rubber spatula like you're making firm scrambled eggs, pulling from the sides toward the middle. Tilt the pan now and then, lifting the egg disk that forms so runny egg slides off, hits the pan, and firms up. After about 2 minutes, when the eggs are slightly shiny on top in spots, but mostly cooked through, spread the mushroom mixture over half of the eggs; sprinkle with cheese. When the eggs are browned on the bottom, fold them over the filling. Slide omelet onto a plate.