Combine the sausage, brown sugar, sage, pepper, and a heavy pinch of salt in a bowl. Work the spices into the sausage and then divide the mixture into 5 portions. Press the portion into super thin patties on squares of parchment paper. Pop open the cinnamon rolls and have these standing by.
Heat a large griddle or skillet over medium high heat. When the griddle is hot add the sausage (parchment side up) and press down like a smash burger. After 2-3 minutes when the sausage is brown, remove the parchment paper and flip the sausage. Cook an additional 2-3 minutes until browned. Remove from the griddle to a plate to keep warm.
Press each cinnamon roll into the hot griddle and cook for 3-4 minutes per side. Remove from the griddle.
Add a little oil to the griddle and crack two eggs together, you may need to work in batches. When the white is just set, break the yolks and add a slice of cheese to each set of cheese. Cover and cook for 1-2 minutes more.
Build the sandwiches by swiping one cinnamon roll with the prepared icing topping with a sausage patty and a fried egg. Top with an additional cinnamon roll and eat immediately.