Heat oil and butter in a heavy skillet over medium heat. Slice chicken breasts in half horizontally to create thin breasts. Sprinkle with seasonings.
Place chicken in the skillet and cook until golden brown on each side, about 4 minutes. Remove chicken to a plate and keep warm.
In the same skillet, melt butter. Add gnocchi and season with salt and pepper. Cook until gnocchi is golden, 6 to 7 minutes. Remove gnocchi to a bowl.
Add broccoli, zucchini, squash, mushrooms, and bell pepper and season with Italian seasoning. Saute until vegetables are crisp and tender, 5 to 7 minutes, depending on the thickness of the vegetables. Add minced garlic during the last 45 seconds; quickly saute until fragrant. Remove vegetables to the same bowl with gnocchi.
Melt 2 teaspoons butter in the same skillet; and cook shallot until softened and fragrant, about 3 minutes. Pour in chicken broth, half-and-half, and heavy cream. Whisk in flour, and cook, whisking to prevent lumps, until thickened, about 3 minutes. Squeeze in lemon juice to taste.
Combine gnocchi and vegetables with cream sauce, and top with cooked chicken.