In a large bowl of a stand mixer fitted with the paddle attachment, mix 1 tbsp. granulated sugar, 1/4 cup warm (110°) water, and yeast. Let stand until softened, about 5 minutes. Add 1 1/4 cups more warm water, salt, 6 tbsp. granulated sugar, shortening, and egg. Mix until blended.
Change to dough hook, gradually add 3 3/4 cups flour, and beat on medium speed until dough is smooth and stretchy and pulls away somewhat from inside of bowl, about 10 minutes (dough will still feel quite wet).
Remove hook from bowl and loosely cover bowl. Let dough rise at room temperature until doubled, about 40 minutes.
In a small bowl, combine 3 tbsp. cinnamon with 1/2 cup granulated sugar. Generously flour a large work surface and scrape dough onto it. Generously sprinkle dough and a rolling pin with flour.
Very gently roll and pat dough into a 10- by 24-in. rectangle, keeping a light coating of flour on dough but trying not to work it in. Sprinkle with cinnamon-sugar mixture and brown sugar, then roll dough into a log starting from a 10-in. side. Cut dough straight down into 8 equal slices. Place slices cut-side up in a greased 10-in. springform pan. Let stand until puffy, about 30 minutes. Preheat oven to 325°.
Set springform pan on rimmed baking pan and bake rolls until golden and a toothpick inserted into the center roll (avoid cinnamon-sugar swirls) comes out clean, about 45 minutes. Let cool in pan on rack, 30 minutes. Meanwhile, make the cinnamon butter: In the bowl of a stand mixer, beat butter with 1 tsp. cinnamon until blended.
Run a knife around inside of pan to release rolls; remove pan rim. With a serrated knife, cut rolls apart and top each with about 1 tbsp. cinnamon butter. Put rolls on a platter and place in warm oven until butter has mostly melted, about 1 minute.
Make ahead: Instead of letting rolls rise on the counter (step 5), immediately cover with a piece of oiled plastic wrap and chill overnight. In the morning, let stand at room temperature until puffy and no longer cold, about 1 1/4 hours. Bake as directed.
Note: Nutritional analysis is per roll.