Whisk together the flour, baking powder, baking soda, salt, and sugar. Beat 1 egg with the buttermilk and melted butter, stir into the dry ingredients until just combined (about 20 strokes), then fold in the maple syrup with 2 to 3 strokes. Rest the batter for 5 minutes.

Cook the 4 sausage patties in a skillet over medium heat for 3 to 4 minutes per side until they reach 160°F (71°C) and are well browned. Set aside.

Wipe the pan and reduce to medium-low. Cook the batter in 1/4 cup portions for 2 minutes per side until bubbles cover the surface and the edges look matte, then 1 minute on the second side. Make 8 cakes total and keep warm under foil.

Add the 1 tbsp neutral oil to the pan over medium heat, fry the remaining egg for 2 to 3 minutes until the white is fully set.

Stack each sandwich: 1 griddle cake, 1 slice American cheese, 1 sausage patty, egg, and a second griddle cake. Press lightly and serve immediately with a drizzle of maple syrup alongside.