Place 1 pound sliced English cucumbers in a colander or large strainer. Sprinkle with ¾ teaspoon of the kosher salt and toss to combine. Let sit for 30 minutes to drain. Meanwhile, toast the breadcrumbs and make the dressing.

Heat 2 teaspoons neutral oil in a small frying pan over medium heat until shimmering. Add ¼ cup panko breadcrumbs and the remaining ¼ teaspoon kosher salt. Cook, stirring occasionally, until the breadcrumbs are light golden brown and crisp, about 2 minutes. Remove the pan from the heat.

Very finely chop 1 ½ anchovy fillets and 1 garlic clove together until paste-like (if using anchovy paste, add ¾ teaspoon directly to the dressing). Transfer to a medium bowl.

Whisk in 1 tablespoon plus 1 teaspoon lemon juice, 1 tablespoon mayonnaise, 1 teaspoon Dijon mustard, ⅛ teaspoon kosher salt, and a few grinds of black pepper 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 cucumbers are ready, pat the slices dry with paper towels (do not rinse). Re-whisk the dressing if needed. Toss the cucumbers in the dressing until evenly coated. Transfer to a large plate, spread out into an even layer, and drizzle any remaining dressing over the top.

Using a vegetable peeler, shave a generous amount of Parmesan cheese (about 1 ½ ounces) over the cucumbers. Sprinkle with the breadcrumbs and a few grinds of black pepper.