In a small bowl, whisk together cold water and cornstarch to create a slurry with no lumps. Add soy sauce, rice wine vinegar, sugar or honey, and grated ginger. Whisk until well combined and set aside.

Slice the carrot thinly on a bias. Slice the mushrooms and cut green beans into thirds. Cut the broccoli into small florets. Dice the onion small.

Heat olive oil and sesame oil in a large skillet over medium-high heat. Add frozen potstickers in a single layer. Fry until the bottoms turn golden brown, about 3-4 minutes. Add 2 tablespoons of water, cover immediately, and cook for an additional 3-4 minutes. Remove potstickers from the skillet and set aside.

In the same skillet, add 1 tablespoon of olive oil. Add sliced mushrooms and diced onion. Cook until mushrooms are browned and have released their moisture, about 3-4 minutes. Remove from the skillet and set aside.

Add 1 teaspoon of sesame oil to the skillet. Add broccoli florets and cook for 3-4 minutes, stirring frequently, until bright green and tender. Remove from the skillet and set aside.

Add 1 tablespoon of olive oil to the skillet. Add sliced green beans and carrots. Cook for 3-4 minutes, stirring often, until vegetables are tender but still crisp. Add the cooked broccoli, mushrooms, and onions back into the skillet. Stir to combine.

Add the cooked potstickers to the skillet with the vegetables. Give the sauce a quick stir to redistribute the cornstarch, then pour it over the mixture. Stir gently to coat everything evenly without breaking the potstickers. Cook for an additional 2-3 minutes until the sauce thickens and becomes glossy.

Transfer to a serving dish and garnish with sesame seeds and sliced green onions. Serve immediately with steamed rice.