Heat the olive oil in a large pot over medium-low heat. Add the shallots and gently cook, stirring occasionally, until soft and translucent, 8-10 minutes.
Add the vegetable broth, peas, ½ tablespoon lime juice, 1 teaspoon salt, and ½ teaspoon pepper. Bring to a boil, then reduce the heat to medium and cook for about 10 minutes.
Remove the pan from the heat and add the basil and Parmigiano-Reggiano. Purée with a stick blender until smooth and creamy. Be patient; peas take a long time to purée and you want it as smooth as possible. (If you don’t have a stick blender, let the soup cool slightly and purée it in batches in a blender. Be sure to crack the lid or remove the center cap to allow the steam to escape and cover with a dish towel so it won't splatter.) Taste and adjust seasoning. I usually add about a ¼ teaspoon more salt, a sprinkling of fresh ground pepper and a squirt of lime. Ladle the soup into bowls and serve hot, or chill and serve cold.
Note: This soup will thicken as it cools. If necessary, add a bit of water to thin to desired consistency.
Freezer-Friendly Instructions: The soup can be frozen for up to 3 months (Wait until you’re reheating the soup to add the basil and Parmigiano-Reggiano). Defrost the soup in the refrigerator for 12 hours and then reheat it on the stovetop over medium heat until hot. (The soup will thicken up while in the freezer; add a bit of water to the soup when reheating to thin, if desired.)