Preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper for easy cleanup.
Toss the cubed squash, sliced carrots, and chopped potatoes with olive oil, salt, and pepper until evenly coated. Spread the vegetables out in a single layer on the prepared baking sheet.
Roast the vegetables in the oven for about 30-35 minutes, until they are tender and slightly caramelized around the edges. The aromas will be sweet and savory as they bake.
While the vegetables roast, prepare the crispy topping by mixing panko breadcrumbs with melted butter, Parmesan cheese, and minced garlic in a small bowl until well combined.
Once the vegetables are roasted and still hot, transfer them to a greased baking dish. Spread the breadcrumb mixture evenly over the top, pressing lightly to help it adhere.
Return the assembled casserole to the oven and bake for an additional 10-12 minutes, until the topping is golden brown and crispy.
Remove from the oven; let it sit for a few minutes to settle, then serve hot. The top should be crispy, and the vegetables soft and flavorful inside.