Preheat the oven to 350 degrees F. Butter an 8-by-8-inch baking dish.
Melt the butter in a large saucepan over medium heat, add the garlic and cook, stirring, for 30 seconds. Add in the potatoes, broccoli stems, cream, thyme, nutmeg, 1/2 teaspoon salt, and a few grinds of pepper. Bring to a boil over medium-high heat and cook, stirring, until the potatoes have absorbed some of the cream and the mixture thickens slightly, 1 to 2 minutes. Remove from the heat and stir in 1 cup of the Gruyere.
Transfer the mixture to the prepared baking dish and shake to distribute the potatoes and broccoli evenly. Top with the remaining 1 cup Gruyere. Cover with foil and bake until the potatoes are fork-tender, about 35 minutes. Uncover and continue to bake until brown and bubbly, about 15 minutes more. Remove from the oven and let cool 10 minutes before serving.
If you don't have a full 3 cups of broccoli stems, add more potatoes.
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