Make the dough: Set aside 1/2 cup of flour. In a large bowl, mix the 2 and 1/4 cups flour, sugar, salt, and yeast together until evenly dispersed. Set aside.
Heat the water, milk, and butter together in the microwave until the butter is melted and the mixture is hot to touch. About 120-130°F. Stir the butter mixture into the flour mixture. Add the egg and only enough of the reserved flour to make a soft dough. Dough will be ready when it gently pulls away from the side of the bowl and has an elastic consistency.
On a lightly floured surface, knead the dough for about 3-4 minutes. Place in a lightly greased bowl and let rest for about 10 minutes.
Meanwhile, make the filling: In a medium bowl, combine the sugar, lemon zest, and vanilla. Stir it around until the mixture becomes sandy. Some juices may be excreted from the lemon zest; that's ok.
After the dough has sat for 10 minutes, roll it out into a 14x8-inch rectangle on a lightly floured surface. Spread the softened butter on top. Sprinkle the lemon sugar filling all over. Starting with the longer side, roll up the dough tightly. Cut into 10-11 even pieces and place in a lightly greased 9-inch round pan. Sprinkle uncooked rolls with additional lemon zest at this point, if desired.
Loosely cover the rolls with aluminum foil and allow to rise in a warm, draft-free place for 60-90 minutes.
After the rolls have doubled in size, preheat the oven to 375°F. Bake for 25-30 minutes until lightly browned. You can cover the rolls with aluminum foil after 15 minutes to avoid heavy browning.
Make the frosting: Right before serving, top your sweet rolls with the cream cheese frosting. Using a handheld or stand mixer fitted with a paddle attachment, beat the cream cheese on medium speed until smooth. Add the confectioners' sugar and lemon juice and beat on medium-high until creamy. Frost warm rolls and sprinkle with additional lemon zest, if desired.