Combine potatoes, 2 cups water, and 1 tablespoon coarse salt in large saucepan. Boil until potatoes are very tender, 15 to 18 minutes. Mash potatoes with water in pan (do not drain water). Add butter and mash until butter is melted. Whisk in eggs, then 1 cup flour; mash until very smooth. Let potatoes stand until barely lukewarm, about 10 minutes.
Meanwhile, pour 1/2 cup warm water into large bowl of stand mixer fitted with paddle attachment; stir in yeast and sugar. Let stand until foamy, about 10 minutes. Add potato mixture to yeast mixture; mix on low speed until well blended, 2 minutes. Mix in 3 cups flour, 1 cup at a time, beating well. Beat until sticky dough forms.
Spread 1/2 cup flour on work surface. Scrape dough out onto floured work surface. Knead until dough is smooth and elastic, adding more flour by tablespoonfuls if dough is very sticky, about 8 minutes.
Coat large bowl with butter. Transfer dough to bowl and turn to coat. Cover bowl with plastic wrap, then kitchen towel. Let dough rise in warm draft-free area until doubled in volume, about 1 hour.
Meanwhile, make filling:.
Mix brown sugar, cinnamon, and flour in medium bowl. Using fork, mix in butter.
Position rack in center of oven and preheat to 425°F Line large rimmed baking sheet with parchment. Turn dough out onto well-floured work surface. Roll out dough to 24x16-inch rectangle. Sprinkle filling evenly over dough. Starting at 1 long side, roll up dough jelly-roll style, enclosing filling. Using large knife dipped in flour, cut roll crosswise into 12 pieces. Transfer rolls to baking sheet, spacing rolls about 3/4 inch apart. Cover baking sheet loosely with plastic wrap. Let rise in warm draft-free area until almost doubled in volume, about 20 minutes (rolls will be very puffy).
Bake cinnamon rolls until golden, about 20 minutes. Cool rolls 10 minutes on baking sheet.
Meanwhile, make glaze:.
Whisk powdered sugar, melted butter, 2 tablespoons milk, vanilla, and coarse salt in small bowl. If glaze is too thick to spread, add more milk by 1/2 teaspoonfuls as needed. Spread glaze over warm rolls.