Peel the kohlrabi and cut in half. Place the kohlrabi halves, cut side down, on a cutting board and cut deeply several times. Bring the broth to a boil. Add the kohlrabi and cover. Simmer over medium heat for 20-30 minutes.
Remove the kohlrabi from the broth. Rinse the basil and pluck the leaves. Set aside a few leaves. Place the remaining basil into the deep cuts on the kohlrabi. Place the kohlrabi, cut side down, in a greased baking dish. Cover with a layer of chedder cheese. Bake for about 20 minutes at 180°C (approximately 350°F).
Peel and finely chop the onion. Peel the garlic. Heat the butter and saute the onion until soft. Press the garlic into the pan. Sprinkle with the flour and saute briefly. Pour in the wine, 200 ml (approximately 3/4 cup) of vegetable broth and the cream. Bring to a boil, stirring constantly. Simmer over low heat for a few minutes. Finely chop the remaining basil and stir into the sauce. Season with salt and pepper. Remove the kohlrabi from the oven and pour the sauce on top.
Cover with cheese and serve hot.