In a large skillet, cook the sausage patties over medium heat until golden brown on each side. Remove to a plate and set aside.
Place the skillet back over medium-low heat and drizzle with a thin layer of vegetable oil.
In a medium bowl, beat together the eggs with the milk until well combined. Season with a pinch of salt and pepper.
Pour 1/4 of the egg mixture into the hot pan in a thin layer and cook just until set. Fold the sides of the egg in towards the center to make a square shape. Remove the egg to a plate and repeat with remaining eggs.
Slice each croissant in half lengthwise and set the top aside.
Place a folded egg over the bottom half of the croissant. Top with a cooked sausage patty (or two if they are small), then a slice of the cheddar cheese.
Cover with a layer of avocado slices, then drizzle with desired amount of the maple dijon mayonnaise.
Finish with a handful of the arugula, add the top of the croissant, and repeat with remaining sandwiches.