Set a fine-mesh sieve over a bowl. In a blender, puree 1/2 pound peas and 1/2 cup water until smooth; spoon into strainer. Repeat with remaining 1/2 pound peas and another 1/2 cup water. Strain puree, pressing firmly against solids with a spoon; discard solids. Stir in 1/2 teaspoon salt and 1/8 teaspoon pepper. Refrigerate until chilled, about 1 hour.
In a small bowl, combine 1 tablespoon water, crème fraîche, lemon juice, and a pinch each of salt and pepper.
Preheat oven to 375°F. Line a large cookie sheet with a reusable nonstick bakeware liner. Drop 6 level tablespoons of Parmesan 3 inches apart onto cookie sheet; spread to form 2-inch rounds. Bake until edges begin to color, 6 to 7 minutes. Transfer crisps (on liner) to a wire rack; cool 2 minutes. Transfer to paper towels to drain. Repeat with remaining Parmesan. Makes 12 crisps.
Place 1/3 cup pea soup into each of six champagne flutes or cordial glasses. Drizzle with crème fraîche mixture and olive oil. Garnish with mint leaves; serve with crisps.