Cook gnocchi according to package directions. Drain in a colander & set aside.
Broccoli
Cut broccoli into florets & place in a ‘steamer’ microwave dish. Cook in the microwave about 1 minute; drain dish & set aside.
Chicken
Season chicken quarters with Italian seasoning, salt & pepper. Melt butter in a skillet over medium heat. Sear chicken on both sides, starting skin side down, until golden brown, about 2-3 minutes per side; drain & set aside.
Asiago Sauce
Preheat oven to 400 F. Butter a rimmed baking dish & set aside.
Melt butter in a skillet over medium heat. Add minced garlic & cook until fragrant, about 1-2 minutes. Gradually whisk in chicken broth & Italian seasoning. Cook, whisking constantly, about 1-2 minutes. Stir in half & half cream & grated asiago cheese until slightly thickened. Adjust seasoning with salt & pepper to taste.
Baking
Arrange chicken quarters in prepared baking pan. Top with cooked gnocchi, broccoli florets & asiago cheese sauce. Roast until completely cooked through, about 25-30 minutes. Serve hot.