Grill the turkey breakfast sausage patty to a golden brown. While the sausage is grilling, prepare the sandwich base.
Heat a non-stick omelet pan over medium-high heat and spray with a non-stick oil spray.
Add the eggs to the pan and cook over easy, not breaking the yolks.
Cut the English muffin in half lengthwise and toast. Spread butter on each cut surface after toasting, if added flavor is preferred. Place a slice of Brie or Smoked Gouda on each half of the English muffin.
Spread a few arugula leaves over the cheese on each half.
Place the hot, grilled, golden brown turkey breakfast sausage over the arugula leaves.