In a medium-sized mixing bowl, add warmed milk, a teaspoon of sugar, and yeast. (You can take a tsp of the sugar out of the 1/2 cup of sugar for the dough.) Set aside. The yeast will start to get foamy.
Using a stand mixer with a paddle attachment, beat together sugar, butter and salt until well combined. Add eggs one at a time and continue beating until eggs are fully incorporated. Turn the mixing speed down to low and slowly add flour until fully incorporated.
Switch out the paddle attachment for the dough hook and pour the yeast-milk mixture into the flour mixture. Using the dough hook, knead the mixture until a soft dough forms (you'll know it's ready when the dough pulls away from the sides).
Grease the inside of a clean medium-sized bowl with vegetable oil. Add the dough to the bowl and flip it over to make sure the dough is oiled on both sides. Cover with a tea towel and let it rest in a warm spot until it doubles in size (approximately 1 hour).
While the dough is rising, make your filling. In a small bowl whisk together sugar, vanilla and lemon zest. Set aside.
Grease a 9×13 inch baking dish.
Once dough has doubled in size, remove it from the bowl and place on a lightly floured surface. Roll the dough into a rectangle that is approximately 1/4 inch thick.
Before sprinkling your filling mix onto the rolled out dough, spread the softened butter onto the dough using the back of a spoon or a spatula. Leave about a half inch around the edges without butter. Sprinkle the filling mix on top of the butter evenly.
Starting with the long edge of the dough, carefully roll it up, tucking as you go. Use a serrated knife and cut the roll into eight even slices, approximately 1.5 inches wide. Place the sliced buns into the greased baking dish. Let the buns again in a warm area, covered, until once again doubled in size.
Preheat oven to 350 degrees F. Place baking dish in the oven (middle rack) and bake for approximately twenty minutes or until golden brown.