Heat the olive oil and vegetable oil in a large nonstick skillet over high heat until they begin to smoke. Add the mushrooms and fry, stirring occasionally, until crisp and browned, about 3 minutes. Remove with a slotted spoon and drain on paper towellined plate. Salt the mushrooms generously. Reserve the oil.
Bring a medium pot of salted water to a boil. Add the peas and blanch for 45 seconds. Drain well.
In a medium bowl, toss the warm sugar snap peas with the shiitake mushrooms and drizzle with some of the reserved oil. Season to taste with salt. Serve warm or at room temperature.