Step 1

Preheat oven to 400°. Butter a shallow 2-qt. baking dish. Melt 2 Tbsp. butter in a medium skillet over medium-high heat. Add breadcrumbs; cook, stirring often, until golden brown, 5–7 minutes. Transfer to a small bowl and toss with ½ cup Parmesan and 1 Tbsp. thyme leaves; season with salt and pepper and set aside.

Step 2

Bring cream, broth, thyme sprigs, and remaining 2 Tbsp. butter to a simmer in a medium saucepan over medium heat; season with salt and pepper. Remove from heat, cover, and let sit 10 minutes; discard thyme. Cover; keep warm.

Step 3

Toss vegetables in a large bowl; season with salt and pepper. Arrange ⅓ of vegetables in dish and top with ½ cup Parmesan. Repeat layers; top with remaining vegetables. Pour cream mixture over; place a piece of parchment paper directly on top of gratin.

Step 4

Bake until vegetables are crisp-tender and cream is thickened, 50–60 minutes. Uncover, top with breadcrumb mixture, and bake until gratin is bubbling and breadcrumbs are brown, 15–20 minutes longer. Let sit 10 minutes before serving.