For the Crust:

1. Prepare the Crust: In a food processor, pulse together flour, sugar, and salt until combined. Add cold butter and pulse until the mixture resembles coarse crumbs. Add the beaten egg and pulse until the dough begins to form.
2. Chill the Dough: Turn the dough out onto a lightly floured surface, knead it a couple of times to bring it together, then wrap it in plastic wrap. Chill in the refrigerator for at least 30 minutes.
3. Roll Out and Cut: Preheat your oven to 350°F (175°C). Roll the dough on a lightly floured surface to about 1/8 inch thick. Use a round cookie cutter or glass to cut out circles and fit them into the cups of a mini muffin tin. Press down gently to form a crust.

For the Filling:

4. Make the Filling: In a mixing bowl, combine corn syrup, granulated sugar, brown sugar, melted butter, eggs, vanilla extract, and salt. Whisk until the mixture is smooth.
5. Assemble the Pies: Place a spoonful of chopped pecans into each crust. Pour the filling mixture over the pecans, filling each crust nearly to the top.
6. Bake: Place the muffin tin in the oven and bake for 15-20 minutes or until the filling is set and the crust is golden. Cover with foil halfway through baking to prevent the edges from getting too dark.
7. Cool and Serve: Remove the pies from the oven and allow them to cool for a few minutes before removing them from the muffin tin. Serve warm or at room temperature.