Heat a large (preferably 12-inch) skillet over medium high heat for 1 minute. Pour in 2 tablespoons of the oil and swirl to coat the pan. As soon as the oil is shimmering -- but not smoking -- add the green beans in an even layer across the pan. Scatter the onion slices on top. Season with salt and pepper and let cook undisturbed for 2 minutes. Add the radicchio and saute, stirring occasionally, until the radicchio has begun to wilt and brown and the green beans are crisp-tender, about another 6 minutes. If the vegetables seem to be cooking too fast or the pan bottom is starting to burn, lower the heat to medium. (If using an electric stovetop, take the pan off the burner momentarily to let the pan cool.) Stir in the garlic, cooking for about another minute to blend in the flavor. Transfer to a serving bowl. Drizzle with the lemon juice and the remaining 1/2 tablespoon oil; toss gently. Season to taste with salt and pepper, garnish with the Parmigiano shavings, and serve immediately.