Combine 1 cup flour, sugar, yeast, and salt in the bowl of a stand mixer.

Combine 1 cup milk and butter in a saucepan over low heat; heat until very warm (120 to 130 degrees F), about 5 minutes.

Beat milk-butter mixture gradually into the flour mixture on low speed. Beat in 1 cup flour and egg. Switch to the dough hook attachment and mix in remaining 2 cups of flour until a soft dough forms.

Turn dough onto a lightly floured surface. Knead until smooth and elastic, shape into a ball, and let rise in a greased bowl until doubled, about 1 hour.

Punch down dough, divide into 2 pieces, and refrigerate for at least 2 hours or overnight.

Combine cream cheese, confectioners' sugar, 1 egg, and lemon juice in a mixing bowl. Beat until fluffy, about 3 minutes.

Roll out each piece of dough into a rectangle, transfer to baking sheets, and spread filling down the center. Cut dough into strips and braid over filling.

Cover the braids and let rise until doubled, about 1 hour.

Preheat oven to 375°F (190°C). Brush tops with milk.