Serve this warm gnocchi dish with a bed of greens topped with sliced tomatoes, shredded fresh basil, a drizzle of balsamic vinegar and olive oil, and a sprinkle of salt and pepper.
In a small saucepan, heat oil, shallots, and butter over medium-low heat until butter melts. Stir in anchovies and crushed red pepper. Cook and stir until heated through, using the spoon to mash anchovies as you stir. Remove from heat. Stir in lemon juice, garlic, thyme, and salt. Let stand at room temperature for 20 minutes.
Meanwhile, cook gnocchi according to package directions, adding the broccolini to water during the last 3 minutes of cooking; drain. Return gnocchi mixture to hot pan; cover to keep warm.
Warm the anchovy sauce over medium heat for 1 minute. Add anchovy sauce to gnocchi mixture along with mozzarella; toss to coat.
To serve, transfer gnocchi mixture to a large serving dish. Top with pine nuts and parsley. Serve immediately.