Bring a large, wide pot of salted water to a boil.
Melt 2 tablespoons of the butter in a large skillet over medium heat. Add the onions, sprinkle with salt and cook, stirring occasionally, until tender, about 4 minutes. Increase the heat to medium-high, add half the radicchio, sprinkle with salt and pepper and cook, stirring and turning, until tender, about 5 minutes. Transfer everything to a medium bowl.
Melt 2 tablespoons of the remaining butter in the skillet; add the remaining radicchio, sprinkle with salt and pepper and cook, stirring and turning, until tender, about 5 minutes. Return the first batch of vegetables to the skillet, stir to combine, remove from the heat and cover to keep warm.
While cooking the radicchio, add the ravioli to the pasta water, stir well and cook until just tender, about 6 minutes. Reserve 1/3 cup of the pasta water, then strain the ravioli in a colander.
Melt the remaining 1 tablespoon butter in the pasta pot over low heat. Add the sage and cook, stirring, about 1 minute. Add the ravioli and toss to coat with sage butter. Add the vegetables and stir well. Add 3 tablespoons of the reserved pasta water and toss to coat. The ravioli should be nicely slicked; if it seems dry, add the remaining water a tablespoon at a time. Season with salt and pepper. Divide the ravioli among four warm bowls and serve.
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