Combine olive oil, garlic, basil, lemon juice, salt and pepper and pour into a Ziploc bag. Add the fish and let marinate for at least 1/2 an hour and up to 2 hours, preferably at room temperature.

Preheat the oven to 400°F.

Pour fish into a baking pan with the marinade. Cover and bake for 15 minutes, uncover and bake for another 10 minutes.

Preheat the oven to 350°F.

Slice peppers in half and core. Arrange halved peppers in a baking pan, skin side up. Drizzle liberally with olive oil and soy sauce. 

Cover and bake for 45 minutes. Uncover and continue baking until slightly charred, about 20 minutes.

Clean and slice mushrooms. Put the mushrooms into a baking pan and drizzle liberally with olive oil and soy sauce. Cover and bake for 45 minutes, mixing occasionally.

Remove the peppers from the oven, cover tightly and let sit for half an hour—this will make sure that the peppers will be easy to peel. After half an hour, carefully remove skins and slice into neat long strips. Reserve 1/2 a cup of pan juices; set aside.

Combine dressing ingredients. Toss with mushrooms and peppers.

Serve warm alongside the tilapia.