Heat the oven to 400°F (200°C). Butter a 2-quart baking dish with 1 tablespoon of the unsalted butter.
Bring 3 quarts water and 1 teaspoon kosher salt to a boil in a large pot. Trim the fennel bulbs, removing stalks and any tough outer layers; halve lengthwise and cut into 1-inch wedges, keeping some core attached so pieces hold together.
Boil the fennel until just tender but not falling apart, 8–10 minutes. Drain well and pat dry to remove excess moisture.
Melt the remaining 1 tablespoon unsalted butter. In a bowl, toss the fresh breadcrumbs with the melted butter, 0.25 teaspoon kosher salt, and half of the Parmigiano-Reggiano (0.5 cup).