Add the flour, nonfat dry milk, sugar and salt to the bowl of a food processor. Pulse several times to combine.
Add the butter cubes to the flour and pulse quickly, about 10-12 times until it resembles coarse crumbs.
Whisk the egg yolk, vanilla extract and ice water together and add to the flour mixture. Pulse quickly, just until the dough starts to come together.
Turn the pastry dough onto a work surface and bring it together in a ball. Flatten into a disc, wrap in plastic wrap and chill for 30 minutes in the refrigerator.
Transfer the pastry to a 9” tart pan with removable bottom. Press the pastry into the bottom and sides of the pan. Prick it all over with the tines of a fork and refrigerate for another 30 minutes.
While the tart shell is chilling, preheat the oven to 375°.
Place a sheet of parchment over the tart shell and weigh it down with pie weights or a pound of dried beans (note: you won’t be able to eat the beans, but you can save them as pie weights for future use).
Bake for 10-12 minutes or until very light golden brown.
FOR THE POACHED APPLES:
In a large saucepan bring the water sugar and honey to a simmer, stirring occasionally until the sugar is dissolved. Add the cinnamon, anise and vanilla bean with seeds. Keep the heat at a low simmer.
While you’re bringing the poaching liquid to a simmer, peel the apples and slice in half vertically. Use a melon baller to remove the seeds and trim away the core with a sharp paring knife.
Transfer the apples to the poaching liquid and simmer for 10-15 minutes or until slightly softened but not mushy. Be sure not to over cook the fruit, otherwise it will fall apart. (Note, you can do this with very firm ripe pears as well).
Use a slotted spoon to transfer the apples to a bowl or storage container. Strain the poaching liquid through a fine mesh strainer over the apples and let cool to room temperature. You can make the apples to this point up to 3 days in advance. Refrigerate until you’re ready to proceed with the recipe.
FOR THE FRANGIPANE:
In a large bowl, cream together the butter and sugar until very light and fluffy. Alternate adding the almond flour and eggs. Start with 1/3 cup of almond flour, mix well and add one egg. Blend thoroughly before adding 1/3 cup almond flour, followed by the egg, etc.
Sprinkle the flour over the mixture and use a spatula to fold it into the frangipane mixture until just blended.
Add the lime zest, almond extract and rum and stir to combine.
Note: you can make the frangipane to this point and refrigerate, covered, until you’re ready to assemble the tart.
ASSEMBLE THE FRANGIPANE TART:
Preheat the oven to 375°.
Transfer the frangipane filling to the tart shell and use an offset spatula to evenly spread the filling into the shell.
Remove the apples from the poaching liquid and pat very dry with paper towels. (Be sure to remove as much liquid from the apples as possible).
Slice the apples vertically from root to tip into 1/4” thick slices. Fan the apple slices in a radial pattern from the center of the tart out to the edge of the shell. Fill in any empty spaces with additional apple slices.
Using two hands, carefully transfer the tart to a baking sheet and place into the oven. Place the apple frangipane tart into the oven and bake for 40-50 minutes or until the frangipane filling is light brown and set.
Remove from the oven and cool to room temperature. When the tart has cooled, add a tablespoon or two of powdered sugar to a mesh sieve and sprinkle the top of the tart with the sugar. Serve.