In a skillet, cook the sausage over medium-high heat, crumbling it with a spoon, until mostly brown but still a little pink, about 5 minutes. Drain on a paper towel; reserve the sausage fat separately.
Butter both sides of the bread with the butter. Arrange half of the bread in 1 layer in a 9-by-13-inch baking dish. Top with half the sausage, followed by half the Gruyere. Repeat with another layer of the remaining bread, sausage and cheese.
In a bowl, whisk together the half-and-half, eggs, 2 teaspoons of the salt and some pepper. Pour the mixture over the layers of bread, sausage and cheese. Cover in plastic wrap and weigh down the strata with a couple sacks of rice/beans in a zipper top bag to compress it. Refrigerate overnight.
The morning of baking, remove the plastic wrap and weights. Preheat the oven to 325 degrees F.
Toss the broccoli with the reserved sausage fat, remaining 1 teaspoon salt and some pepper. Dot the top of the strata with the broccoli florets so the broccoli tops are still visible (to look like trees) but the stalks are submerged in the custard. Bake until the edges pull away from the dish and the center is puffed, about 45 minutes. Let rest for 5 minutes. Right before serving, sprinkle with the Parmesan to mimic snow on trees.
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