Preheat the oven to 350°F (175°C).
Prepare the tart crust: In a large bowl, combine 1 1/4 cups all-purpose flour, 1/4 cup granulated sugar, and 1/2 teaspoon salt.
Add 1/2 cup cold unsalted butter cubes to the flour mixture. Using a pastry cutter or your fingers, cut the butter into the flour until the mixture resembles coarse crumbs.
In a small bowl, whisk together 1 large egg yolk and 2 tablespoons ice water.
Add the egg yolk mixture to the flour and butter mixture. Stir until the dough begins to come together. If needed, add a little more ice water, 1 teaspoon at a time, until the dough holds together.
Shape the dough into a disk, wrap in plastic wrap, and refrigerate for 15 minutes.
Roll out the chilled dough on a lightly floured surface to fit a 9-inch tart pan with a removable bottom.
Press the dough into the tart pan and trim the edges. Prick the bottom with a fork.
Line the crust with parchment paper and fill with pie weights or dried beans. Bake for 15 minutes.
Remove the parchment and weights, and bake for an additional 5 minutes until lightly golden. Remove from oven and set aside.
Prepare the banana cake filling: In a large bowl, cream together 1/2 cup softened unsalted butter and 1/2 cup granulated sugar until light and fluffy.
Add 2 large eggs, one at a time, beating well after each addition.
Mix in 1 teaspoon vanilla extract and 1 cup mashed ripe bananas.
In a separate bowl, whisk together 1 1/4 cups all-purpose flour, 1 teaspoon baking powder, 1/4 teaspoon baking soda, 1/4 teaspoon ground cinnamon, and 1/4 teaspoon salt.
Gradually add the dry ingredients to the banana mixture, alternating with 1/2 cup sour cream, beginning and ending with the dry ingredients. Mix until just combined.
Pour the banana cake batter into the pre-baked tart crust and smooth the top.
Bake the tart for 25 minutes or until a toothpick inserted into the center comes out clean.
Remove from oven and allow to cool completely on a wire rack.
Prepare the banana frosting: In a medium bowl, beat 1/2 cup softened unsalted butter until creamy.
Gradually add 1 1/2 cups powdered sugar, beating until smooth.
Mix in 1/4 cup mashed ripe banana and 1 teaspoon vanilla extract until well combined.
Spread the banana frosting evenly over the cooled banana cake tart.
Chill the tart in the refrigerator for at least 30 minutes before serving to set the frosting.