For the pastry: In a stand mixer fitted with the paddle attachment, beat together the shortening, coconut oil, sugar and salt on medium speed until fluffy, about 5 minutes. Add the egg and vanilla extract and mix until combined. Add the flour and mix on low until the dough comes together in a ball, about 2 minutes. Divide the dough into two even pieces, form into discs and wrap in plastic. Refrigerate for at least 30 minutes.
For the filling: In a large bowl, whisk together the tapioca flour, sugar, cardamom and salt. Add the nectarines, lemon juice, brandy and vanilla paste. Toss gently to combine.
To assemble: Position a rack in the center of the oven; preheat the oven to 400 degrees F. Remove one disk of dough from the refrigerator and leave to soften until pliable, 10 to 15 minutes; reserve the other disk for another use.
Roll the softened dough between two sheets of lightly floured parchment paper into a 14-inch round. While still between the parchment, transfer the dough to a large, rimmed baking sheet. Gently peel away the top sheet of parchment.
Spoon the filling into the center of the pastry and spread it toward the edges, leaving a 2- to 2 1/2-inch margin all the way around. Gently lift the edges of the parchment to help fold the dough up and over the edges of the filling--allow the dough to fold over itself, creating a blanket effect, with crimps and ruffles. Peel away the parchment to leave a border of pastry around the exposed center of the filling.
Whisk together the egg and milk, and brush the mixture over the crust. Sprinkle with the turbinado sugar.
Bake the galette, rotating the baking sheet halfway through, until the crust is golden brown, 40 to 50 minutes. Let cool.
For the coconut whipped cream: Open the can of coconut cream and spoon out the thick, hardened cream into a medium bowl; reserve the watery liquid for another purpose. Add the powdered sugar and almond extract to the coconut cream and beat, using an electric mixer on medium-high speed, until fluffy like whipped cream, 2 to 3 minutes.
To serve, spoon a dollop of the whipped cream onto a slice of the galette and garnish with a pinch or two of the toasted sliced almonds.