Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Add butter and stir until melted. Add onion and saute until golden brown, about 5 minutes. Stir in rice and pour in 1 1/4 cup chicken broth or as much as needed to 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.

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Release pressure carefully using the quick-release method according to manufacturer’s instructions, about 5 minutes. Unlock and remove the lid. Add chopped broccoli and stir to combine. If rice looks dry, add remaining 1/4 cup chicken broth. Close 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. Pour in half-and-half, garlic salt, Italian seasoning, pepper, and 3/4 cup Cheddar cheese. Stir to combine.

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 Cheddar cheese. Broil until cheese is golden brown, about 5 minutes.