In a large serving bowl, beat eggs and cheese to combine. Add salt and pepper to taste. Set aside.
Heat the olive oil and garlic in a large saucepan. Once the garlic begins to turn golden, remove from the oil and discard. Stir in the zucchini, lemon zest and lemon juice. Immediately remove from heat.
Meanwhile, bring 5 quarts of salted water to a boil. Cook the pasta according to package instruction. Drain, reserving 2 cups of the pasta water to finish sauce.
In a large serving bowl, combine hot pasta and zucchini with the oil in the pan. Add the egg and cheese. Toss to coat and to gently cook the eggs with the heat of the pasta. Once combined, add about 1/4 cup of the reserved hot pasta water and toss. If the pasta appears too dry, add another splash of pasta water.
Top with dollops of burrata cheese, fresh chives and a drizzle of oil. Serve immediately.