Prep ingredients per the instructions above.
Make Homemade Fries: Place the potato wedges into a large bowl and cover completely with cold water for 20 minutes.
Blanch the fries. If using the oven: Heat the oven to 350°F (176°C). Place fries on a parchment-lined baking sheet. Cook in the oven until tender, about 15 to 20 minutes. If using oil: Heat the oil to 300°F (150°C) in a Dutch oven or deep fryer. Fry fries in small batches until softened, about 3 to 5 minutes. Set aside.
Make Caesar Dressing: Combine Egg Yolks and Mustard: In a bowl, whisk together the egg yolks and Dijon mustard until well combined. Incorporate the minced garlic and anchovy paste into the egg yolk mixture. Add red wine vinegar and lemon juice to the mixture, continuing to whisk. Begin to slowly drizzle in the extra-virgin olive oil while continuously whisking the mixture. Do this in a slow, steady stream to emulsify the dressing. Gradually add the freshly grated Parmesan cheese to the dressing, whisking continuously. Season the dressing with salt and black pepper to taste. Adjust the quantities according to your preference.
Prepare Croutons: Toss sourdough bread cubes in a bit of olive oil, season with salt and pepper to taste, and bake in the oven at 350°F (180°C) until golden and crispy. Set aside.
Season chicken breasts with salt, pepper, oregano, and garlic powder. In a large skillet over medium-high heat, heat remaining oil until it shimmers. Cook chicken, turning occasionally, until both sides have a nice sear, to an internal temp. of 160ºF (71°C). Transfer chicken to a cutting board. Once cool enough to handle, slice chicken into chunks.
Heat the oil to 375°F (190°C). Fry potato wedges until golden brown and crispy. Remove and place on a paper towel to absorb excess oil. Season immediately with salt and pepper, plus any other desired seasonings.
Assemble Wraps: In a bowl mix together the dressing, croutons, chicken, lettuce, and parmesan cheese. Lay out tortillas and spread the mixture into the middle.
Mix together the ingredients for the lemonade.
Roll the wraps tightly and serve with homemade fries and lemonade.