Roll out the dough, and fit it in a 9 inch pie pan. Crimp the edges. Chill while preparing filling.
In a bowl, whisk together the eggs and sugar. Add the syrup, butter, tea, vinegar, and salt. Whisk until smooth. Stir in the nuts.
Place the pie shell on a cookie sheet, and pour the filling into the shell. Bake in a preheated 450 degree F (230 degrees C) oven for 10 minutes. Lower the heat to 350 degrees F (175 degrees C), and bake until the center is still a bit wiggly. It will set completely when cooled, about 25 minutes. Cool, and serve with sweetened whipped cream.