In a large bowl, combine 3 cups very cold water with 1 dozen ice cubes. In a large saucepan, bring 6 cups well-salted water to a boil; add asparagus. Return water to a boil and blanch asparagus 5 minutes (8 to 10 minutes if using thick spears). Remove asparagus with tongs (reserving water in pan) and place in ice water 1 minute; remove from water, dry, and set aside. Bring cooking water back to a boil and blanch peas about 3 minutes. Remove with a slotted spoon and place in ice water 1 minute. Strain and dry well; set aside.
In a medium skillet over medium-high heat, saute shallot in olive oil until translucent, about 5 minutes. Remove shallot to a small bowl and set aside. Add almonds to hot oil and toast 6 to 8 minutes, stirring frequently, being careful not to burn. Add almonds to shallot and mix in lemon zest, juice, peas, and mint leaves.
Arrange asparagus spears on a serving platter (or divided among six salad plates). Season pea and almond mixture with salt and pepper, then spoon over asparagus. Garnish with Parmigiano-Reggiano.