Heat the oven to 425°F.

Place 1 ¼ cups torn bread pieces close together on a rimmed baking sheet. Drizzle with 1 tablespoon neutral oil, season with ⅛ teaspoon kosher salt, and toss lightly to coat. Spread the bread out. Bake, tossing halfway through, until light golden-brown, 5 to 7 minutes total. Transfer the croutons and any crumbs to a plate; reserve the baking sheet.

Immediately place 1 ½ pounds halved Brussels sprouts in a large bowl, drizzle with 1 tablespoon olive oil, and season with ¼ teaspoon kosher salt and a few grinds of black pepper. Toss to combine. Add the sprouts to the still-hot baking sheet (reserve the bowl) and arrange them cut-side down into a single layer.

Roast until the sprouts are knife-tender and the cut sides are well-browned, 20 to 25 minutes. Meanwhile, make the dressing.

Whisk 4 teaspoons mayonnaise, 1 tablespoon lemon juice, ½ teaspoon Dijon mustard, ½ teaspoon anchovy paste, ¼ teaspoon garlic powder, ⅛ teaspoon kosher salt, and a few grinds of black pepper together in the now-empty bowl until combined. While whisking constantly, slowly drizzle in ¼ cup neutral oil and whisk until thickened and combined. Whisk in 2 tablespoons finely grated Parmesan cheese.

When the Brussels sprouts are ready, add to the dressing and toss to coat. Taste and season with more kosher salt or black pepper as needed. Transfer to a serving platter, and top with the croutons and more Parmesan cheese.