Our cool peach chiffon pie topped with sugared basil leaves is the stuff of summer dreams -- no baking required.
Pulse cookies in a food processor until finely ground. Add butter and 1/2 teaspoon salt; pulse until mixture has the texture of wet sand. Press into the bottom and sides of a 9-inch pie dish. Freeze until firm, 30 minutes.
Meanwhile, sprinkle gelatin over 1/4 cup cold water; let stand until softened. In a large saucepan, combine peaches, lemon juice, 1/2 cup sugar, basil sprig, 3/4 cup water, and remaining 3/4 teaspoon salt. Bring to a boil, then cover, reduce heat to medium-low, and simmer until peaches are very soft, about 15 minutes. Strain through a fine-mesh sieve, pressing on solids to remove as much liquid as possible (discard solids). Stir in gelatin until dissolved. Refrigerate until thickened slightly, about 1 hour; or place in a bowl over an ice-water bath and stir occasionally until it begins to mound, about 10 minutes.
Meanwhile, beat cream with remaining 1/4 cup sugar and vanilla to soft peaks. Gently fold whipped cream into chilled peach puree. Pour mixture into crust and freeze until firm, at least 4 hours and up to 1 day. Remove from freezer 15 to 20 minutes before serving. Top with peach slices and sugared basil leaves; cut into wedges to serve.