Wash and peel 2 Idaho baking potatoes. Cut them into 3/8-inch cubes, and let them sit in water until you are ready to use them. When ready, pour off water, and blot the potatoes dry with paper towels. Melt 1/2 stick (4 tablespoons) of butter or margarine in a deep skillet over low heat. Turn the heat up a bit, and add the potato cubes. Sprinkle 1 teaspooon salt over the potatoes, and immediately cover them with a lid. Let them cook for about 20 minutes, checking and stirring occasionally, until completely done. (They may brown a bit--that's fine; it just adds to the flavor!) While your potato cubes are cooking, prepare your chopped broccoli and your cheese sauce. Place a 10 oz. box of chopped broccoli into a medium saucepan that has about an inch of rapidly boiling water. Let it cook until tender, but still bright green, about 5 to 8 minutes. As soon as the broccoli is done, pour it through a colander to remove water. Meanwhile, prepare your cheese sauce. Melt 3 tablespoons of butter or margarine in a saucepan. Remove from heat and stir in 1/4 cup flour, 2 cups shredded sharp Cheddar cheese, and 2 to 3 cups milk (2 for thick soup, up to 3 for thinner soup). Sprinkle with 1/2 teaspoon salt, and return to cooktop. Cook over low heat, stirring constantly, until thickened and smooth, about 5 minutes. When the potato cubes are tender, remove them from the cooktop to a workspace where you can assemble your soup. If there is excess fat in the bottom of the skillet of potatoes, use a paper towel to absorb and remove the excess fat. Use a potato masher to mash the potatoes a bit, making sure to leave plenty of lumps of potato. Add your cooked chopped broccoli and your prepared cheese sauce. Stir all together, until well combined. Pour into a soup tureen and cover until ready to serve. Serve while hot, transferring ladles-ful to soup bowls. Very flavorful and delightful to look at--a great comfort food!!!