Place the butter in the bowl of a stand mixer fitted with the paddle. Add the coconut oil. Beat on medium speed for 3 minutes. Using a silicone spatula, scrape the sides of the bowl, then add the sugar and vanilla and cream for 4 minutes more, until the mixture is light and fluffy. Add the egg and beat for 2 minutes more, ensuring the mixture is fully combined.
Meanwhile, using a spoon or silicone spatula, mix the flour, desiccated coconut, baking powder, and salt in a medium bowl to disperse the ingredients evenly.
Stop the mixer and add the dry ingredients all at once. Mix on the lowest speed just until the dry ingredients have been fully incorporated.
Cut two large pieces of acetate in half. Turn the sablé dough out onto a work surface and divide it into two equal pieces. Place one piece on a sheet of the acetate and shape it into a flat rectangle. Top with the second sheet, then gently roll the dough to an even layer that is ⅛ inch (3 millimeters) thick. Repeat with the remaining dough. Place the sablé dough, still in the acetate, on sheet pans and freeze until firm, about 1 hour. The frozen sablé dough can be wrapped in plastic wrap and frozen for up to 2 weeks.
Preheat the oven to 350°F (180°C). Lightly coat two half sheet pans with nonstick spray and line them with parchment paper.
Remove one sheet from the freezer and remove the acetate from one side of the dough. Flip it over onto a cutting board and remove the second piece of acetate. Punch rounds of dough using a 2-inch (50-millimeter) plain round cutter. If the dough cracks, it may be too frozen; allow it to temper for a moment before continuing. As you punch each round, transfer it to one of the prepared pans. Repeat with the second sheet of sablé, placing the rounds on the second prepared pan. The trimmed dough can be brought together, rerolled, and frozen to make more sablés.
Bake the cookies for 3 minutes. To keep the cookies flat, remove each pan from the oven and lightly tap on a heatproof surface to release any air pockets that may have started to form. Return to the oven and repeat with the second sheet. Bake for 7 to 9 minutes more, until golden brown. Let cool to room temperature before filling.
The cooled sablés can be stored in an airtight container at room temperature for up to 1 day.