Heat 1 tablespoon oil in a 10-inch broiler-proof skillet over medium-low heat. Add potato, cover, and cook, stirring occasionally, until golden brown and tender, about 10 minutes. Stir in garlic and all but 1 tablespoon of the scallions; season with salt and pepper and cook 1 minute.
In a large bowl, beat eggs until well combined. Add 1/4 cup each tomato and cheese; stir to combine. Add remaining oil to pan, and pour egg mixture over the potatoes.
Preheat broiler with rack 4 inches from the heat. Meanwhile, cook eggs on the stovetop, lifting the edges to allow uncooked egg to flow underneath, until the center is almost set, 8 to 10 minutes. Sprinkle remaining 1/4 cup cheese over the top, then broil in the oven until set, about 2 minutes.
In a small bowl, make a salsa by combining the remaining tomatoes, scallions, cilantro, and lime juice. Run a metal spatula around the edges of the pan and slide the omelette onto a platter. Serve cut into wedges with salsa.