In a small bowl, combine warm milk, yeast, and 1 tablespoon of the granulated sugar. Let stand for 5-10 minutes, until foamy.
In a large bowl, whisk together the remaining granulated sugar, eggs, and melted butter. Stir in the yeast mixture.
Gradually add the flour and salt, mixing until a soft dough forms. Turn the dough out onto a lightly floured surface and knead for 8-10 minutes, until smooth and elastic.
Place the dough in a greased bowl, cover, and let rise in a warm place for 1-1.5 hours, or until doubled in size.
While the dough is rising, prepare the filling by mixing together the brown sugar and cinnamon in a small bowl. In a separate bowl, cream the softened butter.
Once the dough has risen, punch it down and roll it out into a large rectangle, about 12x18 inches.
Spread the softened butter evenly over the dough, leaving a small border. Sprinkle the cinnamon-sugar mixture over the butter.
Starting from the long edge, tightly roll up the dough. Cut the roll into 12 equal slices.
Place the rolls cut-side up in a greased 9x13 inch baking pan. Cover and let rise for another 30 minutes.
Preheat oven to 375°F (190°C). Bake for 15-20 minutes, or until golden brown.
While the rolls are baking, prepare the glaze by whisking together powdered sugar, milk, and vanilla extract until smooth.
Drizzle the glaze over the warm rolls before serving.
Notes
These rolls are best enjoyed fresh but can be stored in an airtight container at room temperature for up to 2 days.