Preheat oven to 350 degrees F. In small bowl, combine brown sugar, corn syrup, butter, and egg; beat well. Stir in pecans and vanilla; mix well.
Spray a rimmed cookie sheet with nonstick cooking spray. Set the filo shells on the cookie sheet. Fill each with 1-2 teaspoons of the pecan filling.
Bake for 15-18 minutes until the filling is bubbly and just set. Remove from cookie sheet and let cool on wire rack. Be careful here - the brown sugar mixture is very hot and it will burn your fingers if you touch it. Use a spatula and a fork to remove the little tartlets from the cookie sheet. And don't delay - that brown sugar mixture hardens quickly and the tartlets can stick together. Store covered in airtight container. Makes 30 tartlets.