In the bowl of a stand mixer, combine the sugar, salt, and yeast. Add two-thirds of the flour and mix well. Zest the lemon into the flour mixture and create a well in the centre.
In a microwave-safe bowl or jug, heat the water and butter in 30-second increments until warm and the butter is mostly melted. Be careful not to overheat the mixture, as it could kill the yeast.
Crack the eggs into a bowl and whisk them well. Pour the eggs into the well in the flour mixture, followed by the warm water and butter mixture. Stir until combined.
Add the remaining flour and knead the dough on medium-high speed for 10-15 minutes. You’ll know it's ready when it no longer sticks to the slides of the bowl. Another way to check if the dough is ready is to perform the windowpane test. (To perform the windowpane test, pinch off a small piece of dough and stretch it gently. If it's stretchy and translucent, you've kneaded it enough.)
Cover the dough with cling film and a cloth, and let it rise in a warm place for about an hour, or until it has doubled in size.
While the dough is rising, prepare the vanilla filling by mixing the softened butter, sugar, and vanilla in a bowl, until smooth.
Once the dough has risen, punch it down and roll it out into a long rectangle on a well-floured surface. Spread the vanilla butter mixture evenly over the dough, leaving a small space on the edge.
From the long end, roll the dough away from you into a tight roll, sealing the bottom edge by pinching the dough together.
Use unflavored floss or a sharp knife to cut the dough into 8 evenly sized rolls. Place them in a greased 8 or 9-inch cake pan and let them rise again for about 30 minutes.
Bake the rolls at 175 degrees Celsius (350 degrees Fahrenheit) for 30 minutes, or until they are golden brown and no longer doughy. Allow them to cool.
While the cake is cooling, make the glaze by, whisking the icing sugar, milk, and vanilla paste or extract in a large bowl until smooth. Adjust the consistency by adding more milk if needed.
Once the rolls have cooled slightly, spread the glaze evenly over the Torta Di Rosa and enjoy!