Add the pasta to a large pot of boiling salted water. Set a timer and cook the pasta for 3 minutes less than it says on the back of the box. When the timer goes off, add the edamame and carrots and cook for 2 minutes longer. Next add the broccoli, and continue boiling for 1 more minute. Drain the pasta and veggies in a colander in the sink. Rinse well under cold water to stop the cooking. Make sure to drain off all the liquid. Add the pasta mixture to the large bowl along with the shredded cheese and cherry tomatoes.
In a small bowl, whisk together the oil and vinegar. Then whisk in the mayonnaise, sugar, garlic and some salt and pepper. Add the dressing to the salad and toss well. Fold in the green onions. Serve immediately, or cover with plastic wrap and chill for 2 hours before serving.