Sprinkle yeast over warm water; stir to combine. Set aside. Place butter and shortening in a large microwavable bowl, and microwave on HIGH until melted, about 1 minute. Stir to combine. Whisk in lukewarm whole milk and yeast mixture.

Transfer mixture to bowl of a stand mixer fitted with whisk attachment. Sift 3 cups of the flour over mixture. Add evaporated milk, sugar, salt, eggs, and egg yolk. Beat on medium speed until smooth, about 1 minute. Remove whisk attachment; replace with dough hook. Gradually add remaining 3 cups flour; beat on medium speed until dough is smooth, about 2 minutes. Let sit at room temperature 5 minutes; beat on medium-high speed until elastic and very smooth, 10 minutes.

Brush top of dough with 1 tablespoon of the melted butter. Cover bowl with plastic wrap; let rise until doubled in size, about 1 hour. Punch dough down, and re-cover. Let rise until doubled, about 30 more minutes.

While dough rises, prepare desired Filling.
For Apricot/Plum Filling: Place apricots or plums in a heavy saucepan. Cover with water; bring to a boil over medium-high. Reduce heat to medium-low, and simmer until the fruit is soft and tender, about 15 minutes. Remove from heat, and let fruit cool completely in liquid, around 15 minutes. Drain and discard liquid; place in a food processor. Add sugar, melted butter, vanilla, and cinnamon. Process until smooth, about 30 seconds.
For Poppy Seed Filling: Stir together poppy seeds, sugar, and flour in a medium bowl. Heat milk in a heavy saucepan over medium-high until milk just comes to a boil, 5 minutes. Reduce heat to medium-low; add poppy seed mixture to milk. Cook, stirring constantly, until thick, about 1 minute. Remove from heat, and stir in butter and vanilla. Cool completely, about 15 minutes.
For Cream Cheese Filling: Combine cream cheese, sugar, egg yolk, and vanilla or almond extract in bowl of an electric mixer. Beat on medium speed until completely smooth and combined, about 1 minute, stopping to scrape the sides and bottom of bowl as necessary.

Generously flour a work surface. Gently roll dough out to a 1-inch-thick rectangle (about 18 x 14 inches). Using a 2-inch round cutter, cut out circles, and place 1 inch apart on parchment paper-lined baking sheets. Brush tops with 2 tablespoons of the melted butter. Let rise in a warm place until almost doubled in bulk, about 20 minutes.

While dough rises, make the Posypka (Streusel Topping): Use a pastry cutter or your fingers to combine sugar, flour, butter, and vanilla or ground cinnamon to form a crumbly texture. Store in the refrigerator until ready to use.

Use your fingers to make 1 small indentation in center of each dough circle, and fill each with about 1 tablespoon Filling. Sprinkle each kolache with 1 to 2 teaspoons Posypka. Let rise until doubled in size, 20 to 30 minutes.

Preheat oven to 350°F. Position racks in top third and lower third of oven. Bake until golden brown, about 20 minutes, rotating baking sheets between top and bottom racks halfway through the baking time. Remove from oven. Brush with remaining 2 tablespoons melted butter. Transfer kolaches to wire racks. Cool completely, about 30 minutes.