They will, but it will take a long time for them to break down so they can eat them. From my experience their food has to be like mush before they can eat it. I would be concerned with the stink factor of the broccoli.

You could try cutting the larger pieces up and mushing them up using a blender.

Actualy , worms eat very little if any vegitable. They consume the micro organisims that grow and feed on decaying veg/plant material. Moving from fruit and melon that is quickly ingulfed by micro critters and become worm food to course stalks you metioned which are slower to decay,you increasinly need to chop and/or age the food. Most things are helped with the process when frozen then thawed. When temps are 100F+,I might place the frozen food in a hole lined with bedding and cover it with more bedding. If you can stand the sight and odor,the worms wil do ok if food is stored in a container where it spoils and molds. Like you,I put excess fruit and veggies in compost. When that compost nears being finished and I don't need it right then,it is often placed in a bin where it serves as both food and bedding.

I have taken a mix of broccoli , celery , carrots , etc. and run them thru a juicer and put that pulp in a canister for a few days until it molds a bit then I feed it to them. At that point it will smell but after putting it in the bin and and covering it up it does not smell at least not for long as the worms will soon devour it.