Combine the first 11 ingredients in a food processor; process until mixture forms a paste. Spoon about 1 1/2 tablespoons mixture evenly into each mushroom cap. Arrange mushrooms in a 10-inch pie plate.
Open a small metal vegetable steamer; place steamer upside down in a large, deep wok. Add water; bring to a simmer. Wearing oven mitts, carefully place pie plate on top of inverted steamer. Cover and cook 6 minutes or until filling is done. Wearing oven mitts, carefully remove pie plate from wok.
Arrange 1/3 cup cucumber and 1/3 cup carrot around edge of each of 4 plates; arrange 4 stuffed mushrooms in center of each plate.