Heat 1 tablespoon oil in a large pan over medium-high heat. Cook the mushrooms, stirring, until tender, 6 to 8 minutes. Transfer to a bowl and set aside.
Add remaining oil to pan. Cook the edamame, snap peas, and ginger, stirring occasionally, until peas are tender and edamame is bright green, 3 to 5 minutes. Add the red pepper, chestnuts, Sriracha, and mushrooms and cook, stirring, until heated through. Season with salt and pepper. Remove from heat; stir in the toasted sesame seeds and vinegar. Serve hot, or refrigerate and enjoy cold.