In a heavy skillet, melt 2 tablespoons of the butter. Add the onions and cook, stirring often, until the onions are caramelized, about 20 minutes. Set aside.

Warm the oil in a deep skillet over moderate heat. Add the pancetta and cook, stirring frequently, until it is crisp, 10 to 15 minutes.

Add the garlic and green beans and cook, stirring occasionally, for about 5 minutes. Add the caramelized onions and chicken stock and cook over low heat, stirring occasionally, until the green beans are tender, 10 to 15 minutes.  

Add butter and cook for another 5 minutes.  Season with salt and pepper.