Zest and juice 1 of the lemons. Slice the remaining whole lemon into thin rounds, removing the seeds. Set them aside.
In a shallow bowl, add the flour, the Parmigiano-Reggiano cheese, 2 teaspoons of the kosher salt, the cornstarch, and the lemon zest and whisk to combine.
Reserve 4 tablespoons of the flour mixture and set it aside.
In a second shallow bowl, add the eggs and the water and beat to combine.
In a 12-inch skillet over medium heat, add the olive oil until it is shimmering.
Dip the artichokes in the remaining flour mixture and then in the egg wash to coat them completely.
Immediately fry the artichokes in batches in the hot oil, flipping them once, until golden-brown, about 3-4 minutes per side.
Transfer the artichokes to a large platter or baking sheet and immediately season them with the extra salt.
Repeat until all the artichokes are fried and seasoned.
Dredge the chicken pieces into the remaining flour mixture and then in the egg wash so all the pieces are completely coated.
Immediately fry the chicken in batches until the chicken pieces reach an internal temperature of 165 degrees F, about 3-4 minutes per side.
Transfer the chicken pieces to the same large platter with the artichokes and immediately season them with the extra salt.
Repeat until all the chicken pieces are fried and seasoned.
Dip the reserved lemon slices into the remaining flour mixture and then in the egg wash so all the slices are completely coated on both sides.
Add the lemon slices in batches to the hot oil and fry, flipping them once, until golden-brown, about 1 minute per side.
Add the fried lemon slices to the platter with the fried artichokes and the fried chicken and season them lightly with the extra salt.
Repeat until all the lemon slices are fried and seasoned.
Transfer the pan from the heat and drain the oil from it and wipe the pan clean.
In the same pan over medium heat, melt the butter.
Add the reserved flour mixture to the melted butter and cook, while whisking, until it begins to brown.
Add the wine and the reserved lemon juice and whisk constantly to combine.
Add the chicken stock and allow the mixture to bubble and thicken, about 10 minutes.
Season the sauce mixture with the extra salt and the pepper.
Drizzle the sauce over the fried chicken-artichoke-lemon mixture on the platter.
Serve garnished with the parsley and the extra lemon wedges.