For the dough, combine the dry ingredients in a mixing bowl and stir well to mix. Cut the butter into 8 to 10 pieces and add to the dry ingredients. Rub the butter in finely; be careful that the mixture remains cool and powdery. Beat the eggs lightly and stir them into the dough with a fork. Continue stirring until the dough masses together. Remove the dough to a floured surface and knead lightly, until the dough is smooth. Form the dough into a rectangle about 1-inch thick and wrap in plastic. Chill the dough until it is firm, about 1 hour.
Butter a 10 x 15-inch jelly roll pan. Roll the dough to a rectangle about 12 x 17 inches. Roll the dough onto the rolling pin, then unroll it into the pan. Press the dough well into the bottom and sides of the pan and fold the excess dough into the pan. Make a high fluted edge around the edge of the pan. Chill the dough-lined pan while preparing the filling.
Preheat the oven to 350 degrees F. Combine the corn syrup and sugar in a saucepan and mix well. Place on medium heat and stir occasionally, until the syrup comes to a boil. At the boil, add the butter and cream and return to a boil. Simmer the mixture for 3 minutes, stirring occasionally. Remove from heat and stir in the vanilla and rum.
Place the chopped pecans in a heatproof mixing bowl. Pour the cooked syrup over the pecans and stir well to mix. Pour the filling on the dough in the pan and spread it evenly with a metal spatula.
Bake the pastry on the lowest rack of the oven for about 30 minutes. Turn the pan occasionally to make sure that the dough and the filling bake evenly. Remove the pan from the oven and cool it on a rack to room temperature. Wrap the pan in plastic wrap and chill overnight.
To unmold the pastry, run a small, sharp knife between the crust and the sides of the pan. Place the pan on low heat for a few seconds to loosen the pastry. Place a cookie sheet on the top and invert. Remove the pan in which the pastry baked and replace it with a cutting board. Invert, remove the cookie sheet and trim away the edges of the pastry.