1. For the Dough: Sprinkle the yeast over the warm water in the bowl of a standing mixer; stir to dissolve. Add the sugar, eggs, milk, and vanilla; attach the paddle and mix at the lowest speed until well combined. Add 3¼ cups of the flour and the salt, mixing at low speed until the flour is incorporated, about 1 minute. Increase the speed to medium-low and add the butter pieces 1 at a time, beating until incorporated, about 20 seconds after each addition (total mixing time should be about 5 minutes). Replace the paddle with the dough hook and add the remaining 1 cup flour; knead at medium-low speed until soft and smooth, about 5 minutes longer. Increase the speed to medium and knead until the dough tightens up slightly, about 2 minutes longer.
2. Scrape the dough (which will be too soft to pick up with your hands) into a straight-sided lightly oiled plastic container or bowl using a plastic dough scraper. Cover the container tightly with plastic wrap and let the dough rise at warm room temperature until doubled in size, 3 to 4 hours. Press down the dough, replace the plastic, and refrigerate until thoroughly chilled, at least 4 or up to 24 hours. Alternatively, for a quick chill, spread the dough about 1 inch thick on a baking sheet, cover with plastic, and refrigerate until thoroughly chilled, about 2 hours.
3. For the Filling: Meanwhile, beat the cream cheese, sugar, flour, and salt in the bowl of a standing mixer at high speed until smooth, 2 to 4 minutes. Add the lemon zest, egg, and vanilla. Reduce the speed to medium and continue beating, scraping down the sides of the bowl at least once, until incorporated, about 1 minute. Scrape the mixture into a small bowl and chill thoroughly before using. (The filling can be refrigerated in an airtight container up to 3 days.)
4. For the Streusel: Mix the brown and granulated sugars, flour, cinnamon, and salt in a small bowl. Add the butter and toss to coat. Pinch the butter chunks and dry mixture between your fingertips until the mixture is crumbly. Chill thoroughly before using. (The streusel can be refrigerated in an airtight container up to 2 weeks.)
5. For the Icing: Whisk all the ingredients in a medium bowl until smooth. (The icing can be refrigerated in an airtight container up to 1 week. Thin with a few drops of milk before using.)
6. When you are ready to shape the coffeecakes,remove the chilled dough from the refrigerator and turn it out onto a lightly floured work surface, scraping the container sides with a rubber spatula if necessary. Divide the dough in half for 2 cakes.
6a. Working with a half recipe of cold dough at a time, shape the dough into a log about 8 inches long and 2 inches in diameter. Using your outstretched hands, roll the log evenly into a 40-inch rope about 1 inch in diameter.
6b. With your fingers together, gently press the log to flatten slightly into a 1½-inch-wide strip.
6c. Using both hands, twist the rope.
6d. Loosely coil the rope in a spiral pattern, leaving a ¼-inch space between coils. Tuck the end under and pinch to secure. Place the coil on a parchment paper-lined baking sheet and cover with plastic wrap. Repeat with second half of dough.
7. Proof until slightly puffed (they will not increase in volume as dramatically as a leaner bread dough), 1½ to 2 hours. (After this final rise, the unbaked cakes can be refrigerated overnight and baked the next morning.)
8. For the Egg Wash: Beat the egg and cream in a small bowl until combined.
9. Adjust an oven rack to the middle position and heat the oven to 350 degrees F. Working with and baking one coffeecake at a time, brush with egg wash evenly over the exposed dough. Place half of the filling over the center of the top, leaving a 1½-inch border around the perimeter. Sprinkle the top evenly with half of the streusel. Slide the baking sheet onto a second baking sheet to prevent the bottom crust from overbrowning and bake until deep golden brown and/or an instant-read thermometer inserted in the center of the cake reads 190 degrees F, 25 to 30 minutes. Slide the parchment with the coffeecake onto a wire rack and cool at least 20 minutes. Drizzle the cake with half the icing and serve.