In a small bowl, blend together the yeast, 1 teaspoon of sugar, and warm milk. Allow this to sit undisturbed for 10 minutes until foamy.
In the bowl of a stand mixer fitted with a whisk attachment, combine the flour, remaining sugar, and salt. Add in the eggs, lemon zest, vanilla, and softened butter. Pour in the bloomed yeast mixture and mix thoroughly until well incorporated. Replace the whisk with a dough hook and knead on medium speed for 7–10 minutes, or until the dough is smooth, elastic, and begins to pull away from the bowl.
Shape the dough into a ball and place it in a greased large bowl. Cover with a towel or plastic wrap and set in a warm location to rise for 1 to 1 1/2 hours, or until doubled in volume.
While the dough is rising, prepare the filling. Mix the softened butter with brown sugar, white sugar, lemon zest, lemon juice, and salt in a small bowl until well blended.
Grease a 9×13-inch baking dish and set aside. Once the dough has risen, turn it out onto a lightly floured surface, gently deflate, and roll into a rectangle approximately 18 by 12 inches and 1/4 inch thick. Spread the lemon filling evenly across the surface using an offset spatula.
Roll the dough tightly into a log beginning from the long edge closest to you. Trim uneven ends if needed and divide the log into 12 equal pieces using floss or a sharp knife. Arrange the rolls in the greased dish and pour the heavy cream in the gaps between them. Cover and allow to proof in a warm place for about 1 hour or until doubled in size.
Preheat the oven to 350°F about 15 minutes before the rolls have finished proofing. Bake for 20–25 minutes until the tops are lightly golden.
As the rolls bake, prepare the frosting. Beat the softened butter in a medium bowl with an electric mixer on high speed until pale and fluffy. Incorporate the cream cheese and mix until smooth. Gradually sift in the powdered sugar while mixing on low speed. Add lemon zest and juice, then increase speed to high and whip until light and airy.
Let the baked rolls cool for 5 minutes before generously spreading the lemon cream cheese frosting over the top. Serve while warm.