Toss the carrots and squash together in a large bowl with 4 tablespoons of olive oil, 1 teaspoon of salt, and 3/4 teaspoon of allspice. Pour onto a large baking sheet and arrange in an even layer. Roast for 15 minutes.
Meanwhile, toss the peppers, mushrooms, and zucchini with 6 tablespoons of olive oil, 1 1/2 teaspoons of salt, and 1/2 teaspoon of allspice. Place on another large baking sheet and spread into an even layer. Toss the broccoli with 2 tablespoons of olive oil, 1 teaspoon of salt, and 1/4 teaspoon of allspice, then place on a small baking sheet.
After the carrots and squash have roasted for 15 minutes, add the rest of the vegetables to the oven and roast for another 15 minutes. Once cooled, toss them all together and enjoy!