Dissolve yeast and 1 pinch sugar in warm milk in a bowl. Let stand until yeast softens and begins to form a creamy foam, 5 to 10 minutes.
Sift flour into a large bowl; make a well in the centre. Add yeast mixture, butter, salt, 1 teaspoon vanilla extract, egg, and 2 tablespoons sugar. Mix gently until a dough forms.
Transfer to a floured surface and knead until dough becomes pliable and is no longer sticky, about 5 minutes. Transfer to a greased bowl. Cover bowl with a damp kitchen towel and leave in a warm place until doubled, 1 1/2 to 2 hours. Punch down dough, cover, and leave until doubled again, about 1 hour.
Grease a baking sheet with oil.
Divide dough into 4 equal parts. Cut each part into 3 equal pieces. Roll each piece of dough into a ball and arrange on the prepared baking sheet. Flatten each piece lightly and let rise for 20 minutes.
Combine farmer’s cheese, egg, 2 tablespoons sugar, sour cream, and 1 teaspoon vanilla extract in a bowl for the filling.
Use a water glass to press an indention for the filling in the center of each dough piece. Spoon filling into the indention and brush with egg yolk. Repeat with remaining dough pieces and filling. Cover and let rise for 20 to 25 minutes.
Preheat oven to 400 degrees F (200 degrees C).
Bake in the preheated oven until golden, 15 to 20 minutes.
In Russia, vatrushka are made with tvorog, but you can use ricotta. Creme fraiche can be substituted for the sour cream.
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