In a large bowl, toss the carrots and parsnip pieces with the brussels sprouts, butternut squash, shallots, thyme sprigs, and 1/4 cup of the olive oil. Season generously with salt and pepper.

Preheat a gas grill to high heat (about 425 degrees). Place 2 perforated grill pans directly on the grate to heat for about 10 minutes. Divide the vegetables between the grill pans and grill over high heat, stirring and turning occasionally, until they are tender and lightly charred in spots, about 50 minutes. Alternatively, roast the vegetables in a large roasting pan in a 425 degrees oven, stirring them occasionally.

Meanwhile, in a small skillet, heat 1 tablespoon of the olive oil. Add the pine nuts and cook over low heat, stirring constantly, until they are lightly browned, about 2 minutes. Add the butter and garlic and cook, stirring, until the pine nuts are browned and the garlic is golden, about 2 minutes. Let cool, then transfer to a mini processor. Add the grated cheese and the remaining 3 tablespoons of olive oil and pulse to a chunky puree. Season the pine nut pesto with salt.

Transfer the vegetables to a large bowl and toss with the pine nut pesto. Serve hot or warm.

The pine nut pesto can be refrigerated overnight. Bring to room temperature before proceeding.