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 and set aside until needed.

While the pasta is cooking, heat 1 tablespoon butter and oil in a large oven-safe skillet over medium heat. Add garlic and cook until fragrant, about 30 seconds. Add shrimp, 2 tablespoons parsley, and dill; cook until shrimp are bright pink on the outside and the meat is opaque, 1 1/2 to 2 minutes per side. Transfer shrimp to a plate, leaving any cooking juices in the pan.

Melt remaining 2 tablespoons butter in the skillet over medium heat. Whisk in flour until fully combined and starting to turn golden, 30 seconds to 1 minute. Whisk in chicken broth until combined, then repeat with milk. Stir in tomatoes, lemon juice, salt, and pepper; bring to a simmer.

Add 1/2 cup Parmesan and Romano; stir until cheeses begin to melt and sauce is creamy. Add 1 tablespoon mozzarella cheese, then mix in 1/2 of the cooked pasta. Add more pasta if there is enough sauce to coat it. Stir in shrimp, then top with remaining mozzarella, Parmesan, and parsley.

Bake in the preheated oven until golden on top and bubbling around the edges, 8 to 10 minutes.