In a medium bowl, crack the eggs and add the salt and toasted sesame oil. Whisk vigorously with a fork or chopsticks for about 1 minute until the mixture is uniform and slightly frothy.

Heat one tablespoon of vegetable oil in a wok or large non-stick skillet over medium-high heat until shimmering. Pour in the whisked eggs and let them puff up slightly around the edges for 5 seconds.

Gently scramble the eggs by pushing them toward the center of the pan, keeping them in large, soft curds. Remove the eggs from the pan while they are still slightly wet, about 1 minute total, and set them aside on a plate.

Add the remaining tablespoon of vegetable oil to the skillet over medium heat. Add the minced garlic and the white parts of the green onions, stir-frying for 30 seconds until highly fragrant but not browned.

Toss in the tomato wedges and turn the heat up to medium-high. Cook for 3 to 4 minutes, pressing gently on some of the tomatoes with your spatula to release their juices, until they soften and form a rich, saucy base.

Stir in the soy sauce, sugar, and chili oil, mixing well to combine the seasonings with the tomato sauce. Simmer the mixture for 1 minute until it thickens slightly and the tomatoes are perfectly tender.

Return the scrambled eggs to the skillet, breaking them up slightly with your spatula. Toss everything gently for about 30 seconds to allow the fluffy eggs to absorb the spicy, savory tomato sauce.

Turn off the heat and garnish the dish with the green parts of the sliced green onions. Serve immediately over warm steamed white rice for a satisfying, classic home-style dish.