Cook chicken fat and skin in water and begin rendering at a gentle temperature (medium-low to medium).
When skin is lightly browned and plenty of fat has been rendered, add onion. Be careful not to overcook fat—it should remain clear and yellow, not brown.
As soon as onion begins to brown, remove from heat.
Strain through a fine-mesh strainer and reserve both fat (schmaltz) and skins.
Lay out skins to dry in refrigerator for 1 hour. Once dry, pan-fry in a small amount of reserved chicken schmaltz until brown and crispy. Reserve remaining schmaltz for dressing and skins for salad compilation.
Yield: 1 cup
Combine juice, garlic, egg yolk, and anchovies in a blender. Run blender on high, drizzling in warm schmaltz.
Add cheese to finish and season with salt and pepper to taste. (Note: if dressing is too thick, thin it out with water.)
Yield: approximately 1½ cups
Preheat oven to 375 degrees.
Tear bread by hand into coarse pieces. Toss pieces in olive oil, salt, and pepper.
Lay bread out on a baking sheet and bake in a 375-degree oven for approximately 15 minutes. Cool and crush by hand (or grind in food processor) to a very coarse breadcrumb.
Split 3 hearts of romaine and then cut into 1-inch pieces.
In a large mixing bowl, toss romaine, dressing, oil, half the Parmesan, salt, pepper, and 2 tablespoons breadcrumbs.
Poach eggs by bringing water to just under a boil, approximately 180–190 degrees. Crack eggs into 4 coffee cups. Add vinegar to water and gently swirl to form a whirlpool. Drop eggs in center. Set timer for 5 minutes and remove with a slotted spoon.
Note: You can serve eggs immediately or place in ice water for up to 2 days.
Portion salad into 4 bowls. Make a nest for egg and gently place egg in middle of each bowl. Season egg with coarse salt and freshly ground pepper. Finish salad with chicken skins, remaining Parmesan, and breadcrumbs.