Heat olive oil in a large frying pan set over medium heat. Add mushrooms and cook, stirring occasionally, until tender and just starting to brown, about 5 minutes. Season mushrooms with salt and pepper.
Meanwhile, cook edamame according to package instructions, rinse under cold water and remove edamame from the pods. Discard pods.
Add the edamame and cherry tomatoes to the mushroom mixture and stir gently until just heated.
For the dressing, whisk together port wine vinegar (sherry vinegar will also work here), smoked paprika and salt. Slowly pour in olive oil, whisking constantly.
Arrange the mushroom and edamame salad in serving dishes and drizzle with dressing. Serve.