Bring a large pot of well-salted water to a boil over high heat. If the sorrel has big woody stems, remove and discard them, and set aside. Cut the larger leaves into thin strips and leave smaller leaves whole. Put the butter in a large skillet over medium heat, and cook, swirling the pan frequently, until golden brown and smells toasty, about 5 minutes. Immediately take the skillet off the heat and set aside.
Add the gnocchi to the boiling water and stir for about 15 seconds. Boil until just tender and they float back up to the top, about 1 minute.
Put the skillet with the brown butter back over medium-high heat. Reserve about 3/4 cup pasta water, then drain the gnocchi well, shaking off as much water as possible. Add the gnocchi right to the butter-be careful as any water clinging to the gnocchi might spatter in the hot butter. Season with salt and pepper. Toss the gnocchi, coating with the butter, until it is browned and crisp in spots, about 1 minute.
Reduce the heat to medium. Add 1/4 cup pasta water and cook, swirling the pan, another 30 seconds. Now add the sorrel and add some pasta water, a spoonful at a time. Simmer, stirring and turning, until the sorrel just wilts and starts to turn a khaki-green color, and the liquid has thickened into a sauce, about 2 minutes. Ignore the color, because sorrel's bright flavor comes out when it cooks like this. Divide among four bowls and grate some cheese over the top.