Salt eggplant and drain on kitchen towels for 30 minutes; press out excess liquid.
Preheat oven to 375°F.
Set up breading station: Season flour with salt and pepper in a shallow dish; whisk up 2 eggs in shallow dish and season with salt and pepper; combine panko, fine crumbs, salt, pepper, 1/2 cup of cheese and parsley in another dish. Moisten the breadcrumbs with 1 tablespoon olive oil. Coat eggplant first in flour, then egg, then breadcrumbs.
Heat a skillet over medium-high heat with a thin layer of olive oil for shallow frying. Sauté each eggplant slice on each side until golden brown.
Heat skillet with olive oil, 2 turns of the pan; add shallots and garlic, and stir 2 minutes. Wilt in the spinach and baby kale; season with salt and pepper. Douse with lemon juice and remove from heat.
Bring a couple inches of water to a low boil. Add vinegar and poach eggs, two at a time for three minutes. Remove to towel and reserve.
In a small saucepot, melt butter and whisk in flour. Cook a minute or so then whisk in warm milk. Season sauce with salt, pepper and nutmeg to taste; stir in remaining cup grated cheese.
Place 2 eggplant rounds on each plate and top each with loosely folded ham then some greens (shake off any excess liquid from greens with your tongs before you pile them on). Carefully place poached eggs on greens and pour the béchamel evenly over the top of the eggs; garnish with minced chives and serve.
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