Position a rack in upper third of oven; preheat broiler to high.

Heat 1 tablespoon oil in a large cast-iron skillet over medium heat. Add onion and cook, stirring occasionally, until starting to brown, 2 to 4 minutes. Add cauliflower and water. Cover and cook until just tender, about 6 minutes. Add kale, garlic, thyme and 1/4 teaspoon each salt and pepper; cook, stirring often, until the kale is wilted, 2 to 3 minutes.

Whisk eggs, paprika and the remaining 1/4 teaspoon salt and pepper in a large bowl. Add the vegetables to the egg mixture; gently stir to combine. Wipe the pan clean; add the remaining 1 tablespoon oil and heat over medium heat. Pour in the egg mixture and top with cheese. Cover and cook until the edges are set and the bottom is brown, 4 to 5 minutes.

Transfer the pan to the oven and broil until the top of the frittata is just cooked, 2 to 3 minutes.