Preheat grill to 375 degrees, and prep for indirect cooking with a pizza stone set in the center of the grill. (If using the oven set to 400 degrees. See notes.)
Combine the flour, salt, sugar, and cold butter in a food processor. Pulse a few times to break up the butter and get a crumbly mixture. Slowly add in the cold almond milk (or water) one tablespoon at a time, pulsing quickly to combine. I usually add 6-8 tablespoons by the time the mixture feels right and slightly moist.
Place dough onto a sheet of parchment paper with a dusting of flour, flatten with hands and shape into a disk. Then place another sheet on top and roll out to an 11" circle.
For the filling: In a separate large bowl gently mix all of the ingredients together until combined.
Spoon berry mixture onto the center of the dough, leaving a border of about 2 inches. Gently fold up the uncovered border over the edge of the fruit mix and pinch into pleats if necessary, creating a 9" round circle.
Brush the pastry with egg wash, sprinkle just the pastry with the shaved almonds and the turbinado sugar.
Transfer to a pizza stone centered on the grill, close the lid, and cook until the pastry is browned and the filling is thickened (time can vary because the heat fluctuates inside a grill). Keep a close eye on it after 35 minutes. This should be done anywhere from 35-45 minutes, or until crust is golden and the filling is bubbly. Cool for 15 minutes before serving.