Begin with the tart shell. In a stand mixer, beat together the butter and sugar until light and fluffy, about 2 minutes.
Add the flour and salt to the stand mixer and beat again until a sand-like mixture forms. Occasionally scrape the paddle and sides during this process.
Add the egg yolk to the mixer, saving the egg white for later in the recipe. Mix on low until a stiff, yellow dough forms. This will take about 30 seconds.
Move this dough to a work surface and use your hands to form it into a smooth ball. Wrap in plastic wrap and store in the fridge while you prepare the rest of the recipe.
Next make the gooey butter topping. There's no need to clean the stand mixer, just get all of the tart dough out of it and add in the butter and sugar. Beat until well combined and fluffy.
Add in the almond flour and beat again to combine.
Add in the egg, egg white, vanilla, salt and flour and mix on medium until a smooth, fluffy, icing-like texture forms.
Spoon this topping into a small bowl and put it in the fridge to chill.
Let both the tart dough and topping chill while you preheat the oven to 350F.
When the oven is ready, use a rolling pin to roll out the tart dough to about 1' thick. It will be hard to handle if you roll it any thinner, so stop here.
Grab a 9" tart pan or cake tin and place the tart dough inside. Use your hands or the back of a wooden spoon to press the dough all the way to the edges.
When the dough fully covers the bottom of the tart pan, prick it a few times with a fork.
Next spread the gooey butter topping evenly on top, and add the fruit as you like.
Bake at 350F for 35-40 minutes, or until the top of the tart is a very light brown.
Slice and serve!