Prepare the Whiskey Glaze: In a small saucepan, heat 1 tablespoon of canola oil over medium heat. Incorporate the minced garlic and cook until aromatic, stirring for about 1-2 minutes. Add water, pineapple juice, brown sugar, teriyaki sauce, whiskey, and cayenne pepper to the pan. Stir well to combine. Let the mixture come to a simmer, then reduce heat to low and cook for about 10 minutes, allowing the flavors to meld. Slowly whisk in the cornstarch mixture, stirring continuously until the glaze thickens, approximately 2-3 minutes. Remove from heat and set aside.

Prepare and Roast the Vegetables: Preheat your oven to 400°F (200°C). On a large baking sheet, toss the carrots and Brussels sprouts with olive oil, salt, and black pepper. Spread the vegetables out in an even layer and roast in the preheated oven for 25-30 minutes, or until they begin to soften and caramelize, stirring halfway through to ensure even cooking.

Glaze and Finish: Once the vegetables are tender and slightly browned, remove them from the oven. Drizzle the whiskey glaze over the roasted vegetables and sprinkle with fresh thyme. Toss gently to coat the carrots and Brussels sprouts evenly in the glaze. Return the baking sheet to the oven for an additional 5 minutes, allowing the glaze to set and caramelize slightly on the vegetables.

Serve: Transfer the glazed vegetables to a serving dish, garnish with extra thyme if desired, and serve warm. Enjoy the sweet, tangy, and slightly spicy flavors of this special side dish.