Activate yeast: Mix ¾ cup lukewarm milk with ½ teaspoon granulated sugar, 1 Tablespoon all-purpose flour, and 1 ½ teaspoon active dry yeast. Let it sit in a warm place until bubbles form on the surface, about 10-15 minutes.
Knead the yeast dough: Place 2 ⅓ cup all-purpose flour, ¼ cup granulated sugar, 2 egg yolks, and proofed yeast inclusive milk in a mixing bowl. Using a stand mixer with a dough hook, mix for one minute until combined. Add ½ stick unsalted butter (slowly melted, not hot) and ¼ teaspoon salt, then knead on medium speed for 8 minutes, scraping down the sides as needed.
Let the dough rise: Turn the dough onto a floured surface, shape into a ball, and place in a bowl. Cover with a towel or plastic wrap, and let it rise in a warm spot until doubled in size.
Prepare the crumbly streusel topping: In a bowl, combine ¾ cup all-purpose flour, ⅔ stick unsalted butter, ⅓ cup granulated sugar, ¼ teaspoon vanilla extract, and ¼ teaspoon ground cinnamon. Use your fingers to work the mixture into coarse crumbs.
Make the cake: Grease a baking sheet with 2 teaspoons solid fat; no parchment needed. Spread the proofed dough into a thin layer, using your fingers, and create slightly higher edges around the sides.
Cover the dough generously with sliced 2 ½ pounds fresh apricots, cut side up. If the apricots are tart, sprinkle them with about two tablespoons of granulated sugar. Finally, sprinkle the apricots with crumbles.
Bake: Preheat the oven to
360 °F and bake the cake for 35 minutes. Once done, remove it from the oven and let it cool on the baking sheet. Before serving, cut the cake into slices about 3 inches wide. If it isn't sweet enough for your taste, you can dust it with 2 Tablespoons powdered sugar.