Preheat the oven to 350 degrees F (175 degrees C).
Bring a large pot of lightly salted water to a boil. Add penne and cook, stirring occasionally, until tender yet firm to the bite, about 11 minutes. Drain.
Heat 1 tablespoon butter and vegetable oil large oven-safe skillet over medium heat. Add garlic; cook until fragrant, about 30 seconds. Add shrimp, 2 tablespoons parsley, and dill. Cook and stir, flipping shrimp halfway, until pink, 3 to 4 minutes. Move shrimp to a plate and set aside.
Melt the remaining 2 tablespoons butter in the skillet, stirring into the shrimp juices. Add flour; whisk until fully combined and starting to turn golden, 30 seconds to 1 minute. Add chicken broth and milk in that order; whisk to combine. Stir in tomatoes, lemon juice, salt, and pepper. Bring sauce to a simmer.
Stir 1/2 cup Parmesan cheese and Romano cheese into the sauce until cheeses begin to melt and sauce is creamy. Add 1 tablespoon mozzarella cheese. Mix in about 1/2 the cooked penne to start; add more if there is enough sauce to coat. Add the shrimp and stir to combine. Top with the remaining parsley, Parmesan, and mozzarella.
Bake in the preheated oven and bake until top is golden and sauce bubbles around the edges of the skillet, 8 to 10 minutes.