In a food processor, pulse almond flour, monk fruit sweetener, xanthan gum, and sea salt until combined. Add cold butter and pulse until coarse crumbs form. Add egg and vanilla, pulsing just until dough comes together. Divide into two discs (one slightly larger), wrap in plastic, and refrigerate for 1 hour.
Peel chayote squash, cut in half and remove seeds, then slice into thin apple-like wedges. Place in a large saucepan with melted butter and lemon juice. Cook over medium heat for 10-12 minutes, stirring occasionally, until slightly softened but still holding shape.
Sprinkle monk fruit sweetener, coconut flour, cinnamon, nutmeg, cloves, and salt over the softened squash. Stir gently to coat evenly. Add apple extract and stir once more. Remove from heat and let cool for 15 minutes.
Remove larger dough disc from fridge and let soften 5 minutes. Roll between two sheets of parchment into a 12-inch circle. Transfer to a 9-inch pie dish, pressing gently into bottom and up sides. Trim excess, leaving 1/2-inch overhang.
Pour cooled chayote filling into bottom crust and spread evenly. Roll out second dough disc same way. Lay top crust over filling or cut into strips for a lattice design. Pinch edges to seal and crimp decoratively. Cut slits in top crust for steam to escape.
Preheat oven to 350°F. Brush top crust with egg wash. Cover edges with a pie shield or foil. Bake for 45-55 minutes, removing shield for the last 10 minutes. Pie is done when crust is golden brown and filling bubbles at slits.
Remove pie from oven and cool on a wire rack for at least 2 hours before slicing to allow filling to set. Slice into 12 even wedges using a sharp knife.