For the Salsa Verde:
Pulse the herbs, garlic, anchovy and capers together in a food processor to chop them without making a paste. Add the oil, a large squeeze of lemon juice, and a pinch of pepper, and process again. Taste for balance. The garlic will taste sharp – don’t worry, it mellows as it bakes.
For the Gratin:
Heat the oven to 400℉.
Cut the squash on the diagonal into ⅛-inch thick slices. Toss them with 1 teaspoon salt, and let stand in a colander or strainer for at least 15 minutes. The squash will release liquid, so set the colander in your sink or over a bowl.
Meanwhile, place the breadcrumbs in a small bowl. Heat the butter in a small pan over medium heat until it browns and smells nutty. Immediately pour it over the breadcrumbs. Scrape the pan to get every delicious drop. Add a pinch of salt and toss well.
Blot the squash thoroughly dry with a towel. Add the shallots, thyme, salsa verde and a few grinds of pepper, and toss to combine. Add all of the cheese and half of the breadcrumbs, and toss again.
Spread the squash mixture evenly in a 9-X-9-inch (or similar volume) oven safe dish. Top with the remaining breadcrumbs. Bake, uncovered, until the squash is tender and the topping is crisped and golden, about 35 minutes. Let stand 10 minutes or longer before serving.