Preheat your oven on the BROIL setting.
In a medium bowl, whisk together the lemon juice with 1 tablespoon water. Whisk in the olive oil and 1/2 teaspoon kosher salt then add the arugula to the bowl and toss to combine. Set the salad aside.
In a large nonstick sauté pan set over medium-high heat, melt the butter then crack the eggs into the pan and cook them until they reach your desired doneness. Remove the eggs from the heat.
Arrange the four bottom halves of the biscuits on a baking sheet. Place one slice of cheese on each biscuit and broil until melted, about 1 minute, then remove them from the oven.
To assemble the sandwiches, smash a portion of the avocado atop the melted cheese then top the avocado with two strips of bacon, one slice of tomato and a fried egg. Top the egg with the dressed arugula then sandwich the fillings together with the top half of the biscuit. Serve immediately.