In a small bowl, whisk together the vegetable broth, soy sauce, vegetarian oyster-style sauce, rice vinegar, brown sugar, cornstarch and sesame oil until no lumps of cornstarch remain. Set it next to the stove.
Prepare all the vegetables before you turn on the heat, keeping the green onion whites and greens separate.
Heat 1 tablespoon of the neutral oil in a wok or large skillet over high heat until it shimmers. Add the broccoli and carrot and stir-fry for 2 minutes.
Add the 2 tablespoons water, cover and let the vegetables steam for 1 to 2 minutes, until the broccoli is bright green and just tender when pierced with a knife.
Uncover, add the bell pepper, snap peas and green onion whites and stir-fry for 2 minutes, until the pepper softens slightly at the edges but still has crunch.
Push the vegetables to the sides of the pan. Add the remaining 1 tablespoon oil to the center, then the garlic and ginger. Stir them in the oil for about 30 seconds, until fragrant but not browned, then toss everything together.
Whisk the sauce again, since the cornstarch settles, and pour it into the pan. Toss for 1 to 2 minutes, until the sauce bubbles, thickens and turns glossy.
Take the pan off the heat, scatter over the green onion greens and sesame seeds, and serve straight away over rice or noodles.