Heat the oil in a heavy skillet, add the diced mushrooms and saute for 5 minutes, or until their moisture has evaporated and they are somewhat dry. Add the peppers and garlic; cook until translucent. Stir in the green onion and then remove the pan from the heat, transfer the vegetables to a plate and refrigerate until completely cooled.
In a large bowl, combine the cooked vegetable mixture with the turkey, salt, pepper, and basil; mix until thoroughly combined. Divide the mixture into 4-6 ounce portions, flatten to form patties, place on a large platter and refrigerate for at least 1 hour. The burgers are suitable to be grilled, sauteed, or baked until an internal temperature of 160 degrees Fahrenheit. Serve the burgers as a main course or a sandwich. They're delicious topped simply with mayonnaise or pesto and served on crisp lettuce with beautiful summer tomatoes.