Adjust oven rack to lowest position and heat oven to 500 degrees. Toss gnocchi with 2 tablespoons of oil and turn onto a large (12- by 18-inch) rimmed baking sheet. Toss mushrooms with another 2 tablespoons of oil, half the vinegar, garlic, basil, oregano, and a generous sprinkling of salt and pepper and turn onto to rimmed baking sheet. Repeat with tomatoes and remaining vinegar, oil, garlic, basil, oregano, salt, and pepper. Scatter mushrooms over tomatoes. Halve the feta and bury each piece into the tomatoes and mushrooms.