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The half-assed lazy student's approach to this most transcendent of all foods is as follows:

2 potatoes (I have no idea what I used) diced and peeled if you want. Waste not, want not, though
a few handfuls frozen broccoli
enough stock to cover the potatoes
two or three cloves finely chopped garlic
any other herbs you fancy
1-1/2 cup preshredded mature cheddar
a lump of salted butter (if you use unsalted I will hurt you)

Ok, here's how to do it.
Set a pot of water to boil, if you're not using stock. Unwrap a few stock cubes and make sure that the water is boiling before you toss them in. Stir slightly, and add your spices.
While that's heating an infusing, dice the potatoes. I did one per serving but that's just me.
Toss them in and wait about half an hour. Stir occasionally and when they're good and ready, remove 1 cup potato chunks sans the broth and mash well with butter and maybe a hint of milk or cream.
return this to the pot and stir. The liquid should thicken very nicely. As an alternative, or if you just like the flavour, add some canned parmesan cheese (yes, the Kraft variety... Don't look at me like that!)
Once the soup is thickened add in the cheese and maybe a little more butter. It's all to taste. Test to see how gooey and sublime it is.

About a minute or two before serving, add in the broccoli. Stir it up and let it become tender but crisp. Mine took three minutes.

Serve and enjoy with gusto. A good garnish is sour cream or a salad. Homemade white bread with a lashing of butter works too.

Warning: may induce an instant heart attack in susceptible individuals. Eat at your own risk.