In a large bowl whisk, together buttermilk, lemon zest, lemon juice, oil, shallots, thyme, 1 tsp of the salt, and 1 tsp of the cayenne, add the chicken, stirring to coat it with the marinade, and let it marinate, covered and chilled, stirring occasionally for 3 hours or overnight. (I put everything in a large zip lock bag.).
Preheat the oven to 425°F.
In a separate bowl, make the egg wash by mixing together the eggs, water and 1 tbl lemon juice.
In another large bowl combine the cornmeal, bread crumbs, Parmesan, parsley, remaining 1 tsp salt, paprika and the remaining 1/2 tsp cayenne pepper. (I put everything in a large zip lock bag, and coat the pieces two at a time.).
Remove the chicken from the marinade with a slotted spoon, letting the excess marinade drip off, dip it in the egg wash, and dredge it in the cornmeal mixture, shaking off the excess.
Arrange the chicken in a one layer onto a rack and let it dry for 30 minutes.
(The chicken may be prepared up to this point 6 hours in advance and kept covered loosely and chilled.).
Arrange the chicken, skin side up, in one layer in a lightly oiled jelly-roll pan, drizzling the butter over it, and bake it in the middle of the oven for 35-45 minutes, or until it is done and crisp.