To make the pesto, combine spinach, broccoli and basil in the bowl of a food processor; season with salt and pepper, to taste. With the motor running, add chicken stock in a slow stream until emulsified; set aside.
In a large pot of boiling salted water, cook pasta according to package instructions; drain well.
Heat 1 tablespoon olive oil in a large skillet over medium high heat. Add tomatoes, and cook, stirring occasionally, until the tomatoes begin to burst and have softened, about 5 minutes; season with salt and pepper, to taste. Set aside.
Heat remaining 2 tablespoons olive oil in the skillet. Add onion, and cook, stirring frequently, until translucent, about 2-3 minutes. Stir in garlic and Italian seasoning until fragrant, about 1 minute.
Whisk in flour until lightly browned, about 1 minute.
Gradually whisk in chicken stock. Bring to a boil; reduce heat and simmer, stirring occasionally, until reduced and slightly thickened, about 4 minutes.
Stir in pesto and heavy cream until heated through, about 2-3 minutes.
Stir in pasta until well combined, about 2 minutes; season with salt and pepper, to taste. Remove from heat; stir in mozzarella and Parmesan until melted, about 1 minute.