In a bowl, combine the peppers and onion and toss with 1 tablespoon olive oil and salt and pepper to taste. Roast for about 30 minutes, until peppers and onions are soft, and their edges slightly blackened. Set aside.
Meanwhile, clean and slice the portobello mushrooms into 1/4-inch strips. Heat 1 tablespoon of olive oil in a frying pan over medium heat and saute until soft. Set aside.
Lightly grease a 9-by-12-inch pan with olive oil. Place one sheet of phyllo on the bottom of the pan and, dipping into the remaining 1/4 cup of olive oil, brush lightly with oil. Repeat with four more sheets, brushing each with olive oil.
Combine the onions, peppers, mushrooms and spinach, and stir well. Spread in an even layer over the phyllo, and sprinkle all of the thyme and cheese over the top. Add another sheet of phyllo and brush lightly with olive oil. Repeat process, brushing each sheet of phyllo with olive oil, until all of the phyllo has been used.
Cover the pan of phyllo with tin foil, and bake for 20 minutes. Remove the foil the last 3 to 4 minutes so it becomes very lightly browned. Watch carefully, as phyllo browns very quickly.
Let the lasagna rest about 5 minutes before serving.