Heat the 1 tablespoon olive oil in a skillet over low heat. Saute the prosciutto, stirring, until crispy, 7 to 10 minutes. Transfer to a paper-towel-lined plate to drain.
Place the zucchini and fennel in a colander and sprinkle lightly with salt. Toss to coat evenly. Set aside for 5 minutes to allow the vegetables to soften and become pliable, then transfer to a large bowl.
Whisk the 1/4 cup olive oil, fennel fronds, mustard, garlic, lemon zest and juice, and some salt and pepper in a bowl. Dress the vegetables. Serve topped with the pecorino and sprinkled with chives.
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