Think of this as an elevated take on an all-American favorite: Our fluffy, mousse-like filling, which tastes of sweet apricots, is spooned into a nutty, press-in cookie crust, drizzled with fruit puree, and sprinkled with pistachios.
Bring apricots, 3/4 cup water, 3/4 cup sugar, and the salt to a boil in a medium saucepan. Cover, reduce heat, and simmer until apricots are very soft, about 10 minutes. Remove from heat, and let cool for 20 minutes.
Puree in a blender. Strain through a fine sieve into a bowl. (You should have 3 cups puree; reserve 1/2 cup.)
Sprinkle gelatin over 1/3 cup cold water, and let stand until softened, about 5 minutes. Heat 2 1/2 cups apricot puree in medium saucepan over medium-high heat. Whisk softened gelatin into puree, and stir until gelatin dissolves.
Prepare an ice-water bath. Whisk together the yolks and 1/2 cup sugar. Whisk in 1/3 of the apricot-gelatin mixture, then pour back into pot. Cook over medium-high heat, stirring constantly, until thickened, 2 to 3 minutes. Pour through sieve into a bowl set in ice-water bath. Whisk until just beginning to gel, 5 minutes.
Whisk whites until soft peaks form. Gradually add remaining 1/4 cup sugar, and whisk until stiff peaks form, about 2 minutes. Whisk 1/3 of the whites into apricot-gelatin mixture. Gently fold in remaining whites. Return bowl to ice-water bath; let cool, stirring, until mixture is thick enough to mound, 3 to 5 minutes. Spoon into crust (it will mound quite high). Refrigerate at least 2 hours or up to overnight. Spoon reserved 1/2 cup apricot puree on top and garnish with pistachios.