Whirl orange juice, soy sauce, vinegar, honey, miso, and shallot in a blender. With blender running, pour in 3 tbsp. vegetable oil and the sesame oil and whirl 10 seconds. Set vinaigrette aside.
Heat remaining vegetable oil in a large frying pan over high heat. Add shiitakes and salt and cook, stirring, until mushrooms are browned, about 3 minutes. Let cool.
In a large bowl, gently toss lettuce and beet greens with 2 tbsp. vinaigrette. Top with shiitakes, edamame, green onions, and sesame seeds. Serve remaining vinaigrette on the side.
Note: Nutritional analysis is per serving.