Combine the flour, salt, and yeast in the bowl of your stand mixer fitted with the hook attachment. In a separate bowl or cup, stir the warm milk and sugar together until the sugar is dissolved.
Turn the mixer onto medium-low speed and begin pouring in the warm milk, followed by the butter, and lastly the egg and vanilla. Turn the speed up to medium to continue bringing all of the ingredients together. Knead the dough at medium speed for 6-7 minutes, adding an additional tablespoon of flour at a time if needed. The dough will be very sticky to start, but will become less so as you build up the gluten. Continue kneading until the dough is very elastic and pulls cleanly off the sides of the bowl. Then, let the dough rest for 5-7 minutes.
While the dough is resting, mix the slightly melted butter, brown sugar, and cinnamon together in a small bowl to prepare the filling. After resting, transfer the dough to a floured surface and roll it out into a 12x14 inch rectangle (with the long end facing you). It doesn't have to be a perfect measurement! Spread the filling in an even layer over the dough.
Starting from the long edge, begin rolling the dough away from you into a tight roll. Pinch the ends to seal the roll and cut it into 11 equal sized pieces.
Line a baking sheet with parchment paper and arrange the rolls into five rows with a 1-2-3-4-1 pattern to form a tree. Make sure to space the rolls 1.5 inches apart to give them room to rise and spread. Cover and let the rolls rise for 30-40 minutes, or until they've grown to at least 1.5x in size.
While the rolls are rising, preheat the oven to 350°F. Bake the rolls for 20-23 minutes, or until they just start to brown.
Make the frosting. Using a stand or hand mixer, beat the butter and salt together for around 30 seconds. Then, add the remaining ingredients and until beating on medium-high speed until fluffy. Frost the rolls and then top them with some holiday sprinkles. Enjoy!