Pour 1/4 cup warm milk and yeast into a small bowl; let rest 5 minutes to soften yeast. Heat butter and 3/4 cup milk until warm (100° to 110°F); butter does need to completely melt.
Pour warm milk and butter mixture into large mixer bowl with 1/2 cup sugar, salt and vanilla. Add eggs, mashed potatoes, 2 cups flour and yeast mixture. Beat on medium speed for 3 minutes. Gradually add enough flour to make a soft dough. Turn out onto a lightly floured surface and knead until smooth and elastic, about 8 to 10 minutes. Place in a greased bowl, turning once to coat. Cover and let rise in a warm, draft-free place, about 1 hour, until doubled in bulk.
To make Caramel Syrup: Combine butter, brown sugar, corn syrup and cinnamon in a heavy saucepan over low heat. Cook until syrup is just starting to bubble around the edges of the pan, about 15 minutes. Remove from heat and pour an equal amount of hot syrup into two greased 9-inch square pans. Sprinkle toasted pecans over syrup and gently stir to coat the pecans. Set aside.
Punch dough down and divide in half. Roll out onto a lightly floured surface with a rolling pin to a 12 x 10-inch rectangle. Spread with half of the softened butter. Combine brown sugar and cinnamon and sprinkle half of the filling evenly over dough. Roll up jellyroll style. Slice into 9 rolls, placing cut side down, in pan, on top of Caramel Syrup. Repeat with remaining dough. Cover and let rolls rise in warm, draft-free place until doubled in size, about 45 minutes.
Bake in a preheated 350°F oven for 20 to 25 minutes, until golden. Remove from oven and let stand 2 minutes. Invert rolls onto serving platter or parchment paper.