Whisk the soy sauce, rice vinegar, and sugar together in a small bowl until the sugar is dissolved.

Whisk the vegetable broth, soy sauce, and cornstarch in a separate bowl until no lumps of starch remain.

Combine the garlic, ginger, pepper flakes, and a teaspoon of oil in a small bowl. Heat a tablespoon of oil in a 12-inch nonstick skillet over high heat until the oil shimmers. Add half of the mushrooms and cook without stirring for 2 to 3 minutes until the bottoms are deeply browned. Flip the mushrooms, reduce the heat to medium, and cook for about 5 minutes until the second side is browned and the mushrooms are tender. Transfer the first batch to a bowl and repeat the process with another tablespoon of oil and the remaining mushrooms.

Return all the cooked mushrooms to the skillet and pour in the glaze. Cook over medium-high heat, stirring frequently, for 1 to 2 minutes until the liquid reduces to a thick, glossy coating. Transfer the mushrooms back to their bowl.

Wipe the skillet clean with paper towels, add the remaining oil, and return to high heat. Add the snow peas and carrots and cook, stirring occasionally, for about 5 minutes until they are bright and crisp-tender. Clear the center of the skillet, add the garlic-ginger mixture, and mash it into the pan surface. Cook until fragrant, about 30 seconds, then stir it into the vegetables.

Return the mushrooms to the skillet. Give the sauce a quick whisk to recombine, then add it to the pan. Cook, stirring constantly, for 1 to 2 minutes until the sauce thickens and binds the vegetables together. Serve immediately while the snow peas are still vibrant.