Heat the oven to 450°F.
In a large bowl, toss the Brussels sprouts with 1½ tablespoons of the olive oil, the salt, and black pepper to taste. Spread the sprouts out on a rimmed baking sheet and roast, tossing them occasionally, until they are crispy and dark gold, 15 to 20 minutes.
Meanwhile, heat a 10- to 12-inch skillet over medium-high heat. Add the cumin seeds to the dry skillet and toast until fragrant, 1 to 2 minutes. Pour the seeds into a large bowl to stop the cooking.
Add 1½ tablespoons of the olive oil to the hot skillet and let it heat until it is shimmering. Add the halloumi cubes and fry, turning them occasionally, until they are golden on all sides, about 5 minutes. Transfer the halloumi to the bowl containing the toasted cumin seeds.
Add the Brussels sprouts to the halloumi. Add the lemon juice, remaining 1 tablespoon olive oil, and Turkish red pepper to taste. Transfer the mixture to a serving platter and top it with additional lemon juice, olive oil, and Turkish red pepper before serving.