Bring 3 cups water to a boil. Remove from heat, add mint, reserving a few sprigs for garnish, and let steep for about 2 hours, until the liquid is very flavorful and slightly green. Strain liquid, discarding mint, and transfer to two ice trays, or an 8-by-8-inch baking pan. Place in the freezer, and freeze until solid.
Meanwhile, melt butter in a medium saucepan set over medium-low heat. Add chopped onion, salt, and pepper; cook, stirring often, until onions are soft and translucent, 8 to 10 minutes.
Add stock, and bring to a boil. Reduce heat, and let simmer 1 hour. The liquid will be reduced by about half. Add peas, and cook about 4 minutes, until peas are just tender. Remove from heat, and transfer to a bowl set in ice water. Stir until cold.
Transfer soup to the jar of a blender; process until smooth. Adjust consistency with more stock. Chill until ready to serve.
Remove frozen mint ice cubes from the freezer, and place them in the jar of a blender (if using an 8-by-8-inch baking pan, cut ice into chunks). Process the mint ice cubes until they are crushed.
Place soup in bowls topped with a spoonful of crushed mint ice; garnish with mint sprigs, and serve immediately.