Cut leeks in half lengthwise, then thinly slice crosswise. Wash leek strips in a bowl of cold water, agitating them, then lift out and pat dry.
Using 2 large skillets or wide, heavy pots, combine 2 1/2 tablespoons oil and 2 1/2 tablespoons butter in each skillet. Heat over medium heat until butter is melted. Divide leeks among skillets and season generously with salt and pepper. Cook, stirring occasionally and reducing heat if necessary, until leeks are very tender and lightly golden, about 40 minutes.
Meanwhile, spread bread cubes in single layers on baking sheets. Bake cubes, rotating pans once halfway through, until dried and lightly golden, 25 to 30 minutes. Let cool on wire racks.
If using giblets, chop neck into 3 to 4 pieces. Heat 2 tablespoons of the remaining butter in a small skillet over medium heat. Add neck pieces, heart and gizzard; season with salt and pepper. Cook, turning pieces occasionally, until giblets begin to brown, about 10 minutes, then transfer to a cutting board. Add remaining tablespoon butter and liver to skillet; season with salt and pepper and cook over medium-high heat for 2 minutes; turn and cook 30 seconds more. Transfer liver to cutting board. Pick meat from neck, discarding the bones. Finely chop liver, heart and gizzards.
Increase oven temperature to 350°.
Combine giblets, leeks and bread in a bowl. Add sage and remaining 1/4 cup oil. Season with salt and pepper and stir together well, then add Brodo and stir to combine. Transfer to a 17- x 12-inch baking dish (or two smaller dishes) and bake until hot and deep golden brown, about 40 minutes.