Make yeast starter: Heat ½ cup milk to 110-115°F, then add ½ teaspoon granulated sugar, 1 Tablespoon all-purpose flour, and 2 teaspoons active dry yeast. Stir and let sit in a warm place for 10-15 minutes until bubbles and foam appear, indicating the yeast is activated.
Make the yeast dough: Put 3 cups all-purpose flour and ⅓ cup granulated sugar in a mixing bowl, then add 2 egg yolks, activated yeast, and ¼ cup milk, lukewarm. Mix with a dough hook for about a minute until roughly combined. Add melted (not hot) ½ stick unsalted butter and ¼ teaspoon salt. Mix on medium speed for 8-10 minutes, occasionally scraping the bowl with a spatula.
Turn the kneaded dough onto a floured work surface using a small flexible spatula. If the dough is too sticky, lightly dust it with flour and fold it twice. Pull the ends from the sides and join them over the center to form a ball.
Place the dough, seam side down, in a large bowl. Cover with plastic wrap or a towel and let it rise in a warm place until doubled in size, about 1-1.5 hours.
Make the cheese pudding filling: While the dough rises, prepare the cheese filling. Heat 1 cup milk in a saucepan until lukewarm. Mix ½ cup of this warm milk with 2 Tablespoon cornstarch, whisking to remove lumps. Bring the remaining milk to a boil, then remove from heat and stir in the cornstarch mixture. Return to the stove and cook on medium heat, stirring constantly, until it bubbles. Set the cream aside to cool, stirring occasionally to prevent a skin from forming.
When the cream is lukewarm, blend in 11 ounces farmers cheese and 1 egg yolk with a hand mixer fitted with a whisk attachement. Add 1 ⅓ cups powdered sugar and ¼ teaspoon vanilla to taste. Leave the cheese mixture at room temperature until ready to fill the Kolache.
Shape the kolache: Turn the dough out onto a lightly floured surface and divide it into 12 parts. I weighed the dough on a kitchen scale, and the whole ball had about 25 oz (700 g). Divide the total weight by twelve to get approximately a 2 oz (60 g) piece for each kolach.
Pull the edges of each dough piece to the center and roll into smooth balls. Place on a parchment-lined baking sheet, seam side down, with space between them. Cover with a tea towel and let rest for 15 minutes to rise slightly.
Wrap a cup (3 ½ to 4 inches in diameter) with a clean dish towel, dip in flour, and press into the center of each dough ball to create a depression with a rim around.
Beat 1 egg with a fork in a bowl. Brush the egg wash around the edges of each Kolach.
Place 1 ½ to 2 tablespoons of cheese filling in the center of each Kolach, spreading it evenly. Add a teaspoon of strawberry jam on top of the cheese filling.
Preheat the oven to
360 °F. Bake the Kolache on the middle rack for 20 minutes. Remove from the oven and let cool on the baking sheet.