Toast each muffin half lightly. Heat your broiler. Grate the cheese; it doesn't have to be finely grated, but it makes a difference if it's not freshly grated - don't use the pre-shredded cheese.
Melt the butter in a small non-stick saucepan, and add the flour to make a roux. Off the heat, whisk in the English mustard, and the milk and cider. Put the pan back on the heat and keep whisking or stirring until you have a thick white sauce (slightly yellow from the mustard). This is a paste more than a sauce and there will not be much of it.
Add the grated cheese, stirring until it melts and then take it off the heat and divide the sauce between the four lightly toasted muffin halves, spreading each one to cover. Immediately place under the heated broiler until they blister on top and begin to brown in spots. Watch them carefully!