Make the Sweet Yeast Dough:
Mix two cups (nine ounces) flour, the sugar, yeast, and salt in a medium bowl with a rubber spatula. Meanwhile, in a small saucepan or in the microwave, combine the milk and the butter and heat until the butter is melted. Remove from the heat, add the water, and let rest a minute until just warm (120 to 130°F [49 to 54°C]). Stir in the vanilla extract. Pour the milk and melted butter into the flour and mix with a rubber spatula until the flour is evenly moistened. Beat in the eggs one at a time.
Pour the milk mixture over the flour-yeast mixture and, using a rubber spatula, mix until the dry ingredients are evenly moistened. Attach the bowl to the mixer, and fit the mixer with the paddle attachment. With the mixer on low speed, add the eggs, one at a time, mixing after each addition just until incorporated. Stop the mixer, add 1/2 cup (2 1/4 ounces) of the remaining flour, and resume mixing on low speed until the dough is smooth, 30 to 45 seconds. Add 2 more tablespoons flour and mix on medium speed until the dough is smooth, soft, and slightly sticky, about 45 seconds.
Lightly flour a work surface and knead the dough gently until smooth and no longer sticky, about one minute. Add an additional 1-2 tablespoons of flour only if the dough is too sticky to work with. Place the dough in a large bowl, cover it with plastic wrap, and let it rise in a warm place (about 70°F [21°C]) for 45-60 minutes or until doubled in size. An indentation made with your finger should keep its shape.
Meanwhile, make the lemon sugar filling. Mix the sugar, lemon zest, and orange zest. It'll draw out the citrus oils and make the sugar sandy and fragrant. Center a rack in the oven and preheat the oven to 350°F (175°C). Grease a 9x5 loaf pan.
Gently deflate the dough with your hand. Flour a work surface and roll the dough into a 20 by 12 rectangle. Use a pastry brush to spread the melted butter evenly and liberally over the dough. Use a pizza cutter to cut the dough crosswise in five strips, each about 12 by 4. Sprinkle 1 1/2 tablespoons of the lemon sugar over the first buttered rectangle. Top it with a second rectangle, sprinkling that one with 1 1/2 tablespoons of lemon sugar as well. Continue to top with rectangles and sprinkle, so you have a stack of five 12 by 4 rectangles, all buttered and topped with lemon sugar.
Slice this new stack crosswise, through all five layers, into 6 equal rectangles. Carefully transfer these strips of dough into the loaf pan, cut edges up, side by side. it might be a little roomy, but the bread will rise and expand after baking. Loosely cover the pan with plastic wrap and let the dough rise in a warm place (70 °F [21°C]) until puffy and almost doubled in size, 30 to 50 minutes. When you gently press the dough with your finger, the indentation should stay.
Bake the loaf until the top is golden brown, 30 to 35 minutes. Transfer to a wire rack and let cool in the pan for 10 to 15 minutes.
Meanwhile, make the cream cheese icing. Beat the cream cheese and powdered sugar in a medium bowl with a wooden spoon until smooth, then add the milk and lemon juice. Stir until creamy and smooth. Flip the loaf over onto a cooling rack, then flip onto another rack so that it's right side up. Spread the top of the warm cake with the cream cheese icing, using a pastry brush to fill in all the cracks.