The night before: In the bowl of a stand mixer fitted with a dough hook, combine the yeast, ½ teaspoon of granulated sugar, and ¼ cup of warm water (about 110-115 degrees, or just warm to the touch). Stir to combine, then set aside for 5-10 minutes or until the yeast becomes frothy. (If your yeast does not become frothy/foamy, this means that your yeast has died and you will need to start over with fresh yeast.)
Add the remaining ¼ cup of granulated sugar, milk, brown sugar, egg, egg yolk, and vanilla to the bowl. Whisk together to combine, then add the flour and salt to the bowl. Mix with the dough hook on medium speed just until the dough comes together. Turn to medium-high and knead for 1-2 minutes.
Add the softened butter to the bowl and continue kneading for another 3-4 minutes. The dough will now be wet and sticky. Turn out the dough onto a well-floured surface and knead 1/3 to ½ cup of flour into the dough with your hands, until the dough is smooth and no longer sticky.
Place dough into a large greased bowl, cover lightly with a towel and set in a warm place to rise until doubled, about 1 ½ to 2 hours.
Once the dough has risen, turn it out onto a floured work surface. Gently knead 4-5 times. Using a floured rolling pin, roll the dough into a 10x20-inch rectangle. If you find that the dough is difficult to roll (it keeps “bouncing” back after you roll it, much like pizza dough), simply cover it lightly with a tea towel and let it rest for about 5 minutes before attempting to roll again.
Spread the berry preserves onto the dough, leaving a 1-inch border along the long edges for rolling. Sprinkle evenly with 1 teaspoon of lemon zest. Rolling from one of the long edges, tightly roll the dough the dough into a cylinder and pinch along the edge to seal. Cut into 8 equal pieces; arrange pieces, cut side up, in a greased 13x9-inch baking dish. Cover with plastic wrap or foil and place in the refrigerator overnight.
The next morning: 1 hour before baking, remove the rolls from the refrigerator and allow rolls to come to room temperature. Near the end of the hour, preheat the oven to 375 degrees. Bake for 25-30 minutes, or until golden brown and bubbling.
While the rolls cool slightly, whisk together the ingredients for the glaze in a medium bowl until smooth. Drizzle over still-warm rolls and serve.