Turn on a multi-functional pressure cooker (such as Instant Pot) and select the Sauté function. Add butter and stir until melted. Add onion and sauté until golden brown, about 5 minutes. Stir in 1 1/4 cups chicken broth and rice; liquid should cover rice. Close and lock the lid. Select Rice setting according to manufacturer's instructions; set timer for 12 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Stir in chopped broccoli and add remaining 1/4 cup chicken broth if rice looks dry. Close the lid and seal the vent. Select Steam and set the timer for 5 minutes.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Add half-and-half, garlic salt, Italian seasoning, and pepper; stir in 3/4 cup Cheddar cheese until combined.
Set an oven rack about 6 inches from the heat source and preheat the oven's broiler. Grease an 8-inch square casserole dish.
Transfer broccoli mixture to the prepared casserole dish and top with remaining 1/4 cup Cheddar cheese. Broil until cheese is golden brown, about 5 minutes.