Preheat the oven to 350F. Begin by making the crust. Add the butter, sugar, vanilla extract, and salt to a bowl. Add the flour and stir until the butter is well coated. Use your fingers to break up the cubes of butter into pea sized pieces. Continue to mix with your hands until a rough dough comes together. It should stick together if squeezed in your fist, but crumble if you run it through your fingers.
Transfer the crust to a 9 or 10 inch tart pan. Use your hands to press the crust up the sides of the pan and evenly across the bottom. Transfer the crust to the oven and bake for 10 minutes. Remove from the oven and set aside to cool slightly.
While the crust bakes and cools, make the filling. Add the butter, sugar, almond extract, and vanilla extract to a medium bowl and whisk until smooth. Add the eggs, one at a time, whisking between each.
Add the almond flour and salt and use a rubber spatula to gently mix until combined, until a smooth thick paste forms.
Transfer the mixture to the crust (it's ok if it's still quite warm) and use the rubber spatula to evenly spread the mixture out to fill up the crust.
Slice all the plums in half, remove the pits, then slice each half into quarters.
Starting in the middle, place the sliced plums on top of the filling in a circle / spiral shape, until the entire tart is covered in plums (see the photo above).
Transfer the tart to the oven to bake for 45-60 minutes, until the filling is no longer jiggly and is just set. Remove from the oven and let the tart cool completely. Enjoy!