Prep the ingredients per the instructions above.
Preheat the oven to 400°F (204°C).
Prep the potatoes: Cut the potatoes into 1/4-inch thick sticks. Place into a large bowl and cover completely with cold water for 20 minutes (see note). Drain well and pat dry with a towel. Place on a foil-lined baking sheet. Cook in the oven, tossing halfway thru until golden brown and crispy.
Make the pickled onions: Heat the vinegar, water, sugar, and salt in a medium saucepan over medium heat. Stir until the sugar and salt dissolve, about 1 minute. Let cool and pour over the onions. Set aside to cool to room temperature, then store the onions in the fridge until ready to serve.
Make the dressing: Place the ingredients in a large bowl and set aside.
For the gravy: Heat the butter in a saucepan over medium heat. Brown the sausage, breaking it up into smaller pieces as it cooks. Stir in the flour and cook while stirring for 1 minute. Whisk in the milk and parsley; bring to a simmer until it starts to become thick and smooth. Season with salt and pepper to taste. Cover with a lid and set aside.
Preheat the broiler over low heat.
Arrange the fries onto a parchment-lined baking sheet. Pour the sausage gravy over the fries. Top with cheese. Place under the broiler until melted.
Make the eggs: Heat the butter in a nonstick pan over medium heat. Crack the eggs into the pan and cook until the white is set. Set aside.
Toss the arugula with dressing and pickled onions.
Mix together the iced tea and lemonade.
Serve the fries topped with fried egg and garnish with parsley. Enjoy with salad and arnold palmer.