This recipe is specifically targeted to folks who are still throwing away their broccoli stems and only eating the florets. You can make this recipe using both parts, or just use the stems. Adding the broccoli to the potatoes makes them moister without adding tons of water or other liquid. Lastly, if you need a way to get your kids to eat their broccoli, this might be it.
Bring a large pot of salted water to a boil and add 1 1/2 lbs. potatoes, quartered or halved depending on their size. Add 1-2 peeled cloves of garlic if you like. Lower the heat and simmer for 10 minutes.
Meanwhile, peel and trim 1 lb. of broccoli stems, then dice them. Add to the pot and boil for another 5 minutes.
Drain the vegetables and transfer them into a large bowl. Add 2 T. olive oil or melted butter and 1 1/2 C. sharp cheddar cheese. Mash until smooth, adding milk or other liquid if necessary to achieve the consistency you like. Add salt and pepper to taste.