Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
In a large bowl, combine the apples with 1/2 cup of the brown sugar, the cornstarch, lemon juice and cinnamon. In another small bowl, combine the remaining 3 tablespoons brown sugar, the oats, butter, flour and salt.
Place the piecrust on the prepared baking sheet. Spoon the apple mixture in the middle of the pie, leaving about a 1-inch border. Fold the crust edges over the filling, pleating occasionally. Spoon the oat mixture over the apples. Brush the crust with the egg and sprinkle with the raw sugar. Bake until the crust is golden brown and the apples are tender, 40 to 45 minutes.
Butter Pie Crust
Pulse the flour, salt and sugar in a food processor. Add about one-third of the butter; process until the butter is combined. Pulse in the remaining butter in 5 or 6 pulses to form coarse crumbs. Add the vinegar; pulse quickly a few times while adding 1/3 cup ice water. Squeeze some dough between your fingers. It should just hold its shape; if it's still crumbly, quickly pulse in more ice water by the tablespoonful. (Do not overwork the dough.)
Turn the dough out onto a large piece of plastic wrap. Wrap the dough, then press and flatten into a thin round. Refrigerate 1 hour.
Roll out the dough into an 11-to-12-inch round, about 1/8 inch thick, on a lightly floured surface. Center over a 9-inch pie plate. Fold in the edges and crimp as desired. Refrigerate at least 30 minutes before using.