Bring a large pot of water to a rolling boil. Add the sweet potato noodles and cook them for 5 to 6 minutes, until they are tender but still springy when you bite one. Start tasting at 4 minutes, because they soften quickly and there is no way to undo it.

Drain the noodles and rinse them under cold running water until they are cool, then snip through the pile three or four times with kitchen scissors so the strands are easier to toss and eat. Toss them with 1 teaspoon of the sesame oil so they do not stick together.

Stir together the soy sauce, brown sugar, sesame oil, and black pepper in a small bowl until the sugar dissolves. Set the sauce aside.

Heat 1 tablespoon of the neutral oil in a large skillet or wok over medium-high heat. Add the tofu in a single layer and cook for 4 to 5 minutes, turning once, until the pieces are golden at the edges. Move the tofu to a plate.

Add the remaining tablespoon of oil to the pan, then cook the onion, carrot, and bell pepper for 3 minutes, stirring often. Add the mushrooms and cook for 2 minutes more, then add the garlic and cook for 30 seconds, until it smells fragrant.

Add the spinach and stir just until it wilts, about 30 seconds, then return the tofu to the pan. Add the noodles and pour over half the sauce. Toss everything together for 1 minute, then add the rest of the sauce and toss again until the noodles are glossy and every strand is coated.

Turn off the heat, scatter the scallions and sesame seeds over the top, and serve warm or at room temperature.