Position a rack in the center of the oven and heat the oven to 375°F.
Put the eggs in a small saucepan and cover with cold water. Bring to a boil, turn off the heat, and cover the pan. Let sit, covered, for 10 minutes. Immediately pour off the hot water and run the eggs under a steady stream of cold water. Peel the eggs right away. Coarsely chop the eggs and set them aside.
Spread the breadcrumbs on a rimmed baking sheet and toast them in the oven until lightly browned, about 5 minutes.
Melt 8 Tbs. of the butter in a heavy-based 10-inch skillet over medium heat. Add the breadcrumbs, paprika, 1 tsp. of the salt, and the pepper and cook, stirring, for about 1 minute, just to meld the flavors. Remove from the heat and stir in the chopped eggs, Parmigiano, parsley, lemon zest, and lemon juice.
In a large pot, bring 1 gallon of water and the remaining 1/2 cup of the salt to a boil. Trim off the bottom of the broccoli stems, cut each broccoli head lengthwise in half, and then cut each half lengthwise into six spears. Add the broccoli to the boiling water and cook until crisp-tender, about 5 minutes.
Drain the broccoli well and arrange in a snug single layer on a rimmed baking sheet. Melt the remaining 8 Tbs. butter in the microwave or in a small saucepan over medium heat. Top the broccoli with the breadcrumb mixture and then drizzle on the melted butter. Bake until the broccoli is heated through and the topping is crisp, about 20 minutes. Transfer the broccoli to a serving platter and then scatter any topping that fell off back over the broccoli.
A day ahead, you can parcook the broccoli and combine the eggs with the parsley, the seasoned breadcrumbs with the Parmigiano, and the lemon juice with the zest. (Store everything separately in the fridge.) Then you can combine the topping ingredients, assemble the dish while the turkey roasts, and bake it while the turkey rests.