Set a steamer rack over a large saucepan of simmering water. Add the potatoes and sliced carrots, cover, and steam until nearly tender, about 9 minutes.
Add the broccoli florets to the steamer, cover, and steam until all vegetables are just tender, about 3 more minutes. Transfer the vegetables to a large bowl and let cool to room temperature.
In a small bowl, whisk together the hoisin sauce and honey. Brush or toss the chopped roasted chicken with the hoisin-honey mixture.
Add the glazed chicken and chopped red onion to the bowl with the cooled vegetables and toss gently to combine.
In a small saucepan, combine the olive oil and sliced garlic and cook over low heat just until the garlic begins to sizzle, about 1 minute; remove from heat.
Whisk the lemon juice and chopped thyme into the warm oil and garlic. Pour the warm dressing over the salad and toss to coat. Season to taste with salt and pepper before serving.