Cook pasta according to package directions, omitting salt and fat. Drain.
While the pasta cooks, shave zucchini and squash into thin ribbons using a vegetable peeler; place in a large bowl. Add tomatoes and next 7 ingredients (through garlic) to bowl; toss to combine. Let stand 15 minutes, stirring occasionally to coat.
Add pasta to vegetables, tossing gently to combine. Top with panko, basil, radishes, and cheese.
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