Sprinkle the yeast and the pinch of sugar over the warm water; stir to dissolve.
Let stand until foamy, about 10 minutes.
Combine the 1/2 cup butter, 1/4 cup sugar, salt, 2 teaspoons of vanilla, milk, and 3 eggs in a bowl with 1 cup of flour and mix well.
Add the yeast mixture and beat for 1 minute. Gradually add the remaining flour to form a soft dough. Turn the dough onto a lightly floured surface and knead until smooth and elastic, adding small amounts of flour as necessary to prevent sticking. Shape the dough into a round, and place it in a greased bowl, turning to coat. Cover with plastic wrap and let the dough rise at room temperature until doubled in size, about 1 1/2 hours.
Beat together the farmers' cheese, 1/3 cup sugar, sour cream, 1 egg, 1 teaspoon vanilla extract, and dried lemon peel in a bowl until smooth. Set the filling aside. Lightly oil a 10-inch fluted tube pan (such as a Bundt®).
Turn the dough onto a lightly floured surface and pat into a 10-inch by 12-inch rectangle.
Brush the dough with the 2 tablespoons of melted butter.
Spread the cheese filling evenly over the dough.
Roll the dough up like a jelly roll, starting from the long end; twist the dough 6 to 8 times to form a rope. Pinch the seams and ends closed and arrange the rope of dough in the greased pan. Cover loosely and let rise 1 hour.
Preheat an oven to 350 degrees F (175 degrees C).
Bake the babka until deep golden brown, 40 to 45 minutes.
Remove from oven and let stand for 5 minutes; invert the babka onto a wire rack and remove the pan. Allow the babka to cool for at least 2 hours before slicing.
