Start by slicing the vegetables. You can either use a mandoline slicer or a sharp knife to slice into round shapes. I try to pick vegetables that are similar in thickness to make a pretty Briam and for the veggies to cook evenly.
To a large mixing bowl, add the sliced veggies and drizzle with olive oil, add garlic, oregano, salt, and pepper. Give everything a toss so that the veggies are well seasoned.
To a 9x13 inch oven-proof dish, add the tomato sauce and ½ cup of water. Then arrange the seasoned vegetables in rows. You can also randomly lay out the vegetables.
If there's any olive oil left in the mixing bowl, pour that over the vegetables along with the thyme sprigs. Tent foil the baking dish, and place in a preheated oven.
Bake at 390°F (200°C) for 30 minutes, check if the potatoes are tender and remove the foil. Roast for 20 more minutes for the liquid to reduce and the vegetables to start browning.
You may want to place the dish under the broiler for 5 minutes to get the veggies to crisp up a bit. Allow to cool for 20 minutes, then serve with feta cheese, crusty bread, and olives.