In a large saucepan, cook the diced bacon over medium heat until very crisp. With a slotted spoon, transfer the bacon to a plate lined with paper towels. Pour the fat out of the skillet and reserve the skillet (don't clean it). Mix the breadcrumbs with the melted butter until well combined; add a pinch of salt and the bacon bits and set aside.

Heat the oven to 350 degrees F. Rub a shallow 2-quarts gratin dish with 1/2 teaspoon of the butter.

Over medium-low heat, melt the remaining 3 tablespoons butter in the pan in which the bacon was cooked. Scrape the browned bits from the bottom of the pan as the butter melts. Add the cabbage and season with about 1/4 teaspoon salt and some pepper. Turn the heat to medium and cook, stirring frequently, until the cabbage is wilted and beginning to brown, 4 to 5 minutes. Add the garlic, stir, and cook until fragrant, about 1 minute. Add the chicken broth and the cream, stir and scrape the pan, and remove from the heat.

Cut the potatoes in half lengthwise and slice the halves thinly. Put the potatoes in a large bowl with a scant 2 teaspoons kosher salt and a few grinds of pepper. Add the cabbage mixture, scraping the pan well. Add the Gruyere and, using a rubber spatula, mix gently and thoroughly. Scrape the mixture into the prepared gratin dish, smoothing and pressing until evenly distributed. Cover with the breadcrumb topping and bake until the topping is well browned and the potatoes pull away from the edges and are just tender enough to poke through with a fork, about 1 hour. Let cool for 15 to 20 minutes before serving.

Leftovers: Transfer any leftovers to a smaller ovenproof dish that's just big enough to hold them, taking care to keep the crust on top. To reheat, drizzle with a little chicken broth, cider, or milk and bake, uncovered, at 350 degrees F until heated through, about 30 minutes.