In a saucepan, heat the chicken stock till hot.
In a large skillet, heat the olive oil over medium heat. Add the onion and sauté until translucent, 8 to 10 minutes. Add the rice and stir with a wooden spoon until toasted, 3 to 4 minutes.
Add about ⅔ cup of stock to the pan, just enough to cover the rice, and stir until the liquid is absorbed. Add the radicchio to the pan and stir to incorporate. Continue adding the stock, 2/3 cup at a time, stirring until the liquid is absorbed and the rice is creamy but al dente, about 18 minutes. You may have stock left over.
Remove the pan from the heat and add the butter and both cheeses. Stir vigorously until all is melted, blended and creamy.
Serve immediately with additional grated Parmigiano.