Preheat the oven to 400 degrees F. Combine the garlic and olive oil in a small saucepan over low heat. Gently simmer, stirring occasionally, until the garlic is lightly golden and crisp, 5 to 6 minutes. Remove from the heat; transfer the garlic to a paper towel-lined plate using a slotted spoon and season with salt. Reserve the garlic oil.
Toss the Brussels sprouts with the reserved garlic oil on a baking sheet and spread in an even layer; season with salt and pepper. Roast until the Brussels sprouts are tender and slightly charred, 30 to 35 minutes.
Meanwhile, combine the vinegar, brown sugar, honey and bay leaf in the saucepan and whisk until combined. Simmer over medium-high heat, whisking occasionally, until slightly syrupy, 10 to 12 minutes. Remove from the heat; let cool until thick and glossy, about 6 minutes. Remove the bay leaf.
Transfer the Brussels sprouts to a platter; spoon the vinegar-honey syrup on top and sprinkle with the fried garlic and shaved cheese just before serving. (The Brussels sprouts can be made up to 2 hours ahead. Serve at room temperature.)