Combine the water, sugar, a 2" long piece of lemon peel, the juice of the lemon, half of the vanilla bean and the rum in a large pan. Bring the water to a boil. Lower the pear halves into the water and reduce the heat to keep the water at a simmer. Simmer until the pears are just tender, about 10-15 minutes. You don't want them to become soft and mushy. Turn off the heat and set the pears aside until the syrup is completely cooled.
Roll the dough to fit a 9" or 10" fluted tart pan. Prick the dough all over with a fork and set it aside while you make the frangipane filling. Preheat the oven to 350 °F
In a mixing bowl combine the sugar, butter and salt. Scrape the seeds out of the other half of the vanilla bean and add them to bowl. Cream the mix together until light and aerated. Scrape the bowl and the beater. Add the eggs and mix to combine. Add the almond flour and rum.
Spread the frangipane evenly into the tart crust. Lift the pears halves out of the syrup, reserving the poaching liquid in the pot. Blot the pears dry with a paper towel. Slice each pear half into 1/4" thick slices across the short side, keeping the half together. Lift a sliced pear with a small spatula and set it into the tart with the stem end towards the center. Arrange the other pear halves evenly spaced on the tart.
Bake on the lowest rack of the oven until the filling is set, about 40 minutes. While the tart is baking return the poaching liquid to the stove and bring it to a boil. Let the liquid boil until it is reduced to about 1 cup of thick syrup. If you have a thermometer cook until about 225 °F. Turn off the heat and set the syrup aside to cool.
Once the tart comes out of the oven allow it to cool for 10 minutes. Lightly brush the reduced poaching syrup over the tart. Cool to room temperature. Best served at room temperature.