Combine breadcrumbs and 2 Tbsp. oil in a medium skillet; toast over medium heat, tossing occasionally, until golden brown and crisp, about 7-10 minutes. Remove from heat and add thyme and ¼ cup Parmesan; season with salt and pepper. Set aside. (These can be made a day in advance. Store in an airtight container at room temperature)

Cook collard greens and kale in a large pot of boiling salted water until tender and bright green, about 3-4 minutes. Drain, transfer to a bowl of ice water, and let cool. Drain again and squeeze dry with paper towels. Coarsely chop greens..

Heat remaining 2 Tbsp. oil in a medium saucepan over medium heat. Add onion and garlic and cook, stirring often, until softened and golden, 5-10 minutes. Transfer to bowl with greens; set aside. Reserve saucepan.

Increase oven temperature to 400°. Melt butter in reserved saucepan over medium heat. Add flour and cook, whisking constantly, until mixture is smooth and very pale brown, about 4 minutes. Gradually whisk in milk, ½-cupful at a time; add nutmeg. Bring to a boil, reduce heat, and simmer, whisking often, until thickened, 5–8 minutes. Whisk in remaining ¾ cup Parmesan. Add this béchamel sauce to collard green, kale and onion mixture and mix to combine; season with salt and pepper.

Transfer collard green mixture to a 10” cast-iron skillet or 9” pie dish and top with breadcrumb mixture. Tear the proscuitto and roll or lay flat over the bread crumbs. Bake until gratin is bubbling, 15–20 minutes. Let cool slightly before serving.