Heat the oven to 350ºF, with a rack in the middle. Line a baking sheet with parchment paper.
Arrange the prosciutto slices on the baking sheet, leaving a little space between them. Bake until they are shriveled and dark red and the fatty edges are golden, 12 to 15 minutes; keep an eye on them to prevent burning. Place the prosciutto “chips” on a wire rack set over a baking sheet to cool; they will crisp as they cool.
Meanwhile, cut the Brussels sprouts lengthwise in half through the stem end. Make a V-shaped incision at the base of each sprout half. Remove and discard any blemished outer leaves.
Heat 1 tablespoon of the butter and 1 tablespoon of the oil in a large skillet, preferably cast-iron, over medium-high heat until the foam subsides. (The pan needs to be superhot, so test it with a sprout leaf to see if it crisps upon contact.) Place about one third of the Brussels sprouts in the pan, arranging them so they are mostly cut side down, with a little space between them. Reduce the heat to medium and cook until the undersides are deeply golden, about 3 minutes. Turn the sprouts over, sprinkle with a pinch of salt, and continue cooking until golden and tender, 3 to 4 minutes more. Using a slotted spoon, transfer the Brussels sprouts to a large serving bowl. Repeat with the remaining butter, oil, and sprouts.
Crumble the prosciutto chips into small pieces and add to the bowl with the Brussels sprouts. Add the lemon zest and lemon juice, then toss to combine. Season with salt to taste. Serve hot.