In a food processor, pulse the flour, salt, and sugar. Add the shortening and butter and pulse until only pea-size pieces remain. Drizzle in 3 tablespoons ice-cold water (or up to 2 more as needed), and pulse until dough looks crumbly but holds together when pinched. Remove; form into a disk. Wrap in plastic and chill for 1 hour or up to overnight.
Preheat oven to 425°. On a lightly floured work surface, roll the dough into an 11-inch round (sprinkle the surface and rolling pin with more flour as needed). Transfer the dough to a standard (not deep) 9-inch pie plate. Gently press it into the plate and trim excess.
Prick the dough with a fork 8–10 times and chill for 10 minutes or until firm. Put a sheet of parchment paper over the crust and fill with dried beans or pie weights.
Bake until lightly golden and beginning to set, about 15 minutes. Remove the pie weights and parchment paper and bake 10–15 minutes more. Remove, transfer to a rack, and let cool.
Using an electric mixer, whip the heavy cream until thickened slightly. Add the confectioners sugar and vanilla and continue whipping until medium peaks form.
Fill the crust with a third of the cream. Top with the banana then half of the remaining cream, spreading to distribute and create a dome shape.
With a star-tipped piping bag, pipe the remaining cream into small spirals in a single layer around the pie's edge (do not cover the crust).
Chill the pie for at least 2 hours or up to 8. Top with the toasted almonds. Pie is best served the day it's made.