In a large skillet, heat the canola and sesame oils over medium-high heat. Add the mushrooms and cook, stirring frequently, until the mushrooms are just beginning to brown, about 3 minutes. Sprinkle the sugar over the mushrooms and then add the wine and soy sauce. Cook, stirring, for 1 minute.
Add the sugar-snap peas, cover, reduce the heat to medium, and cook until the pea pods are just crisp-tender and bright green, 2 to 3 minutes. Serve immediately.