Cut off zucchini ends, then cut each zucchini in half lengthwise. Slice into ¼-inch thick half moons.
Wipe mushrooms clean of any dirt, and remove stems.
Cut caps into ¼-inch thick slices.
Peel and thinly slice shallots.
Peel and mince garlic clove.
Stack basil leaves together, then roll into a tube. Slice thinly into slivers.
Heat 1 tablespoon olive oil in a 12-inch nonstick skillet over medium-high heat until shimmering.
Add zucchini slices and shallots and sauté until tender and browned at the edges, about 15-20 minutes. Season with salt and pepper.
Add garlic and cook for an additional minute, then remove vegetables from pan.
Add another tablespoon of olive oil, then add mushroom slices and cook until they release their liquid and start to brown, about 10 minutes. Season with salt and pepper.
Add zucchini back to pan, and pour in chicken broth.
Let simmer for 5 minutes, until liquid reduces slightly.
While vegetables are cooking, bring a pot of water to boil and add pasta.
Cook to al dente, according to package directions.
Drain pasta, reserving ½ cup pasta water.
Add cooked pasta to skillet and toss until well-combined.
Add pasta water as necessary to coat the pasta.
Taste, and add more salt and pepper if necessary.
Transfer pasta to serving platter, and top with burrata, pulling it apart with your fork.
Drizzle with olive oil and garnish with basil.
Notes
Storage Tips: Refrigerate leftovers in an airtight container for up to 4 days.Expert Tips: Don’t overcrowd your skillet. You might be tempted to cook the zucchini and the mushrooms at the same time, but this will just end up in steaming both of them. You’ll lose the depth of flavor that comes from the browning of each ingredient. Similarly, if your zucchinis are particularly large, you may find you’ll need to cook them in separate batches. Cooking the pasta just to al dente before draining is important, as the pasta will continue to cook when you add it back into the pan with the sauce.