Simmer apricots with 1 1/2 cups water and the honey, covered, until apricots are very soft, 15 minutes; let cool. Take 1 qt. ice cream out of freezer to soften.
Arrange 1 cup apricots over cooled cookie crust. Whirl remaining apricots and liquid in a blender into a thick, pourable purée (add more water if needed). Strain.
Stir softened ice cream until smooth. Spread over apricots in crust, set on a plate, and freeze until firm, about 1 hour. Spoon on all but 2 tbsp. apricot purée. Freeze until purée firms, about 1 1/2 hours.
Let remaining 1 qt. ice cream soften; spread over purée. Top ice cream with small dollops of remaining purée, then swirl with a chopstick. Freeze 2 hours more.
Remove sides of pan and pat almonds firmly into sides of pie. Serve immediately.
*Get super-flavorful dried Blenheim apricots by mail from brfarms.com Find crunchy, very almondy Italian amaretti in well-stocked grocery stores. Or use almond biscotti instead and whirl in 1 tsp. almond extract with the melted butter.
More pie tips:
Let the pie soften for 5 minutes at room temp to make slicing easier.
If you're having trouble freeing your pie from its pan, set it over a bowl of hot water for a couple of minutes and then slide a thin knife between the pan edge and the crust. It should pop right out.
Make ahead: Once the pie is fully frozen through step 4, it keeps for up to 4 days, double-wrapped in plastic wrap. Top with almonds just before serving.