Vegan Ginger Udon Noodle Bowl

2½ cups uncooked whole grain, gluten-free udon noodles
2 cups uncooked edamame
1-inch piece ginger, peeled
1 large carrot
½ cup water, as needed
1 teaspoon maple syrup
2 tablespoons miso paste
¾ cup chopped green onions
¼ teaspoons black pepper (optional)