Chicken-Rice Soup with Broccoli

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1 bunch broccoli (about 1 pound)

3 tablespoons extra-virgin olive oil

1 onion, chopped

3 scallions, white and light green parts roughly chopped, dark green parts thinly sliced

1/2 cup roughly chopped fresh dill leaves, stems roughly chopped and kept separate

Kosher salt and freshly ground pepper

3 cloves garlic, finely chopped

1 teaspoon Italian seasoning

7 cups low-sodium chicken broth

1 pound skinless, boneless chicken breasts (2 to 3)

1/2 cup basmati rice

Grated parmesan cheese, for topping