Puree garlic cloves in the bowl of a food processor. Add sun dried tomatoes and pulse till coarsely crumbled. Add oregano and pulse. Add cheese, 1 tbsp olive oil and balsamic vinegar; pulse till crumbly. Season to taste with salt.
Measure 3/4 cup of crumbles, reserving remainder for another use. Heat a large sauté pan over medium high heat. Add 2 tsp olive oil and heat. Add onions and sauté till limp and just beginning to caramelize. Add zucchini strings and sauté gently until zucchini is bright green, about 4 minutes.
Add the sun dried tomatoes, parmigiano crumbles and toss to coat zucchini completely.
Adjust seasonings to taste with kosher salt. Serve immediately with reserved crumbles sprinkled on top.