Fill a large pot with water and bring it to a boil over high heat.

While the water heats, put the low-sodium soy sauce, dark soy sauce, vegetarian oyster sauce, and brown sugar in a small bowl. Stir until the sugar is mostly dissolved, then set the sauce near the stove.

Take the tofu from its package, pat it very dry with paper towels, and cut it into 1/2-inch thick rectangles or squares.

Cut the broccoli into small florets and thinly slice the stem pieces so they cook quickly. Slice the scallions, keeping the white parts separate from the green parts, and finely chop the garlic.

Add the rice noodles to the boiling water and cook until they are just tender but still a little firm in the center, following the package time as a guide. Stir once or twice so they do not stick together.

Drain the noodles in a colander and rinse them briefly under cool water to stop the cooking. Shake off the extra water and set the noodles aside close to the stove.

Set a large skillet or wok over medium-high heat. Add 1 tablespoon of the oil, then add the tofu in a single layer.

Cook the tofu until browned on the first side, about 3 minutes. Turn the pieces and cook the second side until browned, about 2 to 3 minutes more, then transfer the tofu to a plate.

Add another 1 tablespoon of oil to the same skillet. Add the scallion whites and the garlic, and stir for about 30 seconds, just until fragrant.

Add the broccoli and 2 tablespoons of water to the skillet. Cover and cook for 2 minutes, then uncover and stir until the broccoli is bright green and just tender, about 1 minute more.

Push the broccoli to the sides of the skillet to make an open space in the center. Pour the beaten eggs into that space and cook, stirring gently, until the eggs are softly set into loose pieces.

Add the drained noodles and the remaining 1 tablespoon oil to the skillet. Use tongs or two spatulas to lift and turn the noodles with the broccoli and egg so they start to separate.