Preheat the oven to 400ºF (200ºC) and set a rack to the middle position.
Prepare the vegetables: In a large bowl, toss the potatoes and carrots with 2 1/2 tablespoons olive oil, thyme, rosemary, salt, and pepper.
First roasting: Spread the potatoes and carrots onto a rimmed baking sheet in an even layer. Roast for 20 minutes.
Prepare zucchini: Toss the zucchini pieces in a bowl with the remaining 1/2 tablespoon olive oil and season lightly with salt.
Add to baking sheet: Add the zucchini and minced garlic to the sheet. Toss everything together and spread again into an even layer.
Final roasting: Roast for another 20 minutes until all vegetables are tender and slightly browned.
Serve warm: Garnish with extra fresh herbs if desired and serve immediately.