Bring a pot of salted water to a boil. Cook the gnocchi according to the package directions. Reserve ½ cup of the cooking water, then drain the gnocchi well.

Heat a large skillet over medium-high heat. Add the sausage and cook, breaking it into bite-size pieces, until browned and cooked through. Add the shallots and cook briefly until softened. Transfer the sausage and shallots to a plate.

Add the green beans and a splash of water to the skillet. Cover and cook until hot and tender-crisp, about 3–5 minutes. Transfer to the plate with the sausage.

Reduce the heat to medium. Add the butter and sage leaves. Cook, swirling the pan, until the butter smells nutty and the sage is crisp. Remove the sage leaves to a small plate.

Add the drained gnocchi to the skillet. Pan-fry until golden on several sides, adding a little olive oil if the pan looks dry.

Return the sausage, shallots and green beans to the skillet. Pour in the reserved cooking water and add the Parmesan. Toss until the sauce coats everything.

Season with black pepper and taste before adding salt. Serve topped with the crispy sage and more Parmesan.