Broken Lasagna With Zucchini-Tomato Sauce

Kosher salt

large zucchini (about 1 3/4 pounds), coarsely grated

lasagna noodles (not no-boil), broken into bite-size pieces

unsalted butter

cherry tomatoes (1 cup whole, 1 cup halved)

finely grated lemon zest

Freshly ground pepper

grated parmesan cheese, plus more for topping

small bunch chives, cut into 1-inch pieces