Heat olive oil in a large skillet over medium-high heat. Place the halved Brussels sprouts cut-side down. Sear undisturbed for 4-5 minutes until they have lightly charred edges. Remove from heat.

Boil the cubed golden potatoes and chunky orange carrots in salted water for 6-8 minutes until barely fork-tender. Drain and let steam-dry for 5 minutes.

Melt 4 tbsp butter in a saucepan over medium heat. Whisk in flour and cook for 1-2 minutes. Gradually whisk in the whole milk and heavy cream. Simmer for 3-5 minutes until the white cream sauce is thick and bubbly. Season with salt and pepper.

Preheat oven to 375°F (190°C). Add the potatoes, carrots, and Brussels sprouts to a rustic ceramic baking dish. Pour the thick cream sauce over the vegetables, ensuring they are heavily coated.

Toss the breadcrumbs with 2 tbsp melted butter and sprinkle an even crust over the top. Bake uncovered for 25-30 minutes until golden-brown and bubbly. Let rest for 10 minutes, then garnish sparsely with chopped green parsley flakes.