First, make the dough. Combine the flour and salt in a medium bowl. Set aside.
Mix warm milk and 1 tsp. of sugar in the bowl of a stand mixer. Sprinkle the yeast over the top and let it stand for 5-10 minutes, until foamy. Add the remaining sugar, butter, eggs, and vanilla until well combined. Then, mix in the flour mixture until the dough starts to come together. Using a dough hook (or your hands), knead for 7-10 minutes, until the dough is smooth and slightly tacky. It should be a little sticky but shouldn't leave any dough on your hands. If it's too sticky, add flour 2 Tbs. at a time until it's the right consistency.
Place the dough in an oiled bowl, and cover with a towel or plastic wrap. Place it in a warm, draft-free spot and let it rise until doubled in size, about 1 1/2-2 hours.
While the dough is rising, make the filling. Stir together all of the ingredients. Set aside until ready to use.
Butter a 9x13'' rectangle pan (or a 12'' round pan). Set aside. When the dough is doubled in size, punch it down. Transfer the dough to a clean, lightly floured surface, and roll until you have an 18x12'' rectangle. Spread the filling on the dough, leaving a 1/4'' gap on all sides. Tightly roll the dough length-wise, and pinch the edges to seal. Place the large roll in the freezer for 10-15 minutes (it makes it easier to cut!).
Once the dough is cold, cut the long roll into 10-12 cinnamon rolls and place them evenly in the buttered pan. Cover with a towel or plastic wrap, place it in a warm, draft-free area, and let the rolls rise for another 30-60 minutes, until puffy but not bubbly. Alternatively, skip the second rise, and refrigerate the rolls overnight to bake them the next day.
*Note: To bake them the next day, remove the rolls from the fridge and let them rise at room temperature for 1-2 hours, until puffy.
Preheat the oven to 350° F. Lightly warm 1/4 c. of cream, and brush it over the rolls. Bake until the rolls are a light golden brown, about 18-20 minutes. (I always under-bake them a little, because they will continue cooking once you take them out of the oven). Let them cool for about 5-10 minutes before frosting.
Make the frosting while the rolls rise/bake. Melt the butter in a saucepan over medium heat. Cook, swirling occasionally until the butter is toasty brown and smells nutty. (Be careful at this step as it burns quickly). Pour the butter into a bowl, and let it solidify on the counter or fridge until it's a softened texture.
Combine the brown butter, powdered sugar, vanilla extract, and salt. Add milk 1 Tbs. at a time until it reaches your desired consistency (it should drizzle a bit when poured). Frost the cinnamon rolls, and serve warm.