Chop the carrots diagonally first, then straight. Cut off the ends of the asparagus, and shave their ends.
In a stock pot, bring 1 gallon of salted water to a boil. Add onions, let cook 2 minutes, the. Add carrots and asparagus, cook 2 minutes more. Remove everything with slotted spoon and drain all in a colander. Quickly spread out on a baking sheet and place in refrigerator for 5 to 10 minutes to set color.
Heat 1/4 cup olive oil in a large skillet over medium-high heat. Add the mushrooms in a single layer and cook until well caramelized on the first side, about 4 minutes. Season with salt and pepper. Turn the mushrooms once, and use a slotted spoon to remove the mushrooms onto the plate.
In a roasting pan or skillet over medium-high heat, add remaining 1/2 cup of extra-virgin olive oil, onion, and carrots. Caramelize for 2 minutes, and while those are cooking, cut asparagus into thirds. Season carrots and onions with salt and pepper. Add mushrooms to pan with carrots and onions. Add asparagus and garlic to pan.
Drizzle with more olive oil, if desired, and place pan in the oven. Roast until vegetables are medium brown in color, about 20 minutes. Remove pan from oven and put on stovetop over high heat. Add the fresh thyme, by sprinkling over the top, allowing to crackle, and mixing in.
The vegetables can be cooked and left at room temperature at this point for up to 2 hours. Reheat when needed.