Preheat the oven to 425 degrees F.
Measure 2 cups of flour into a bowl, add the salt and grated cheese and mix lightly with a fork. Pour the milk into a measuring cup to come up to 1-cup mark and then crack in the egg and add the oil. Beat to combine, then pour into the dry ingredients, forking to mix as you go. You may, at the end, feel the dough's either too dry or too damp: add either more milk or more flour and fork together again until you've got a soft dough that's not too sticky to be rolled out.
Break the dough into 2 pieces and roll 1 piece on a lightly floured surface. Scone dough is a dream to work with; in fact, I find it deeply pleasurable. Just roll as clumsily and heavy handedly as you like: no harm will come to it. You want a thin, but not exaggeratedly so, rectangle. A square wouldn't be the end of the world either, so don't start getting out the geometry set: this is the roughest of instructions.
Cut the dough into approximately 1 3/4-inch strips, and then cut each strip at approximately 2 1/2-inch intervals so that you end up with a collection of small, raggedy oblongs (I just cut each strip as I go, but it's probably more efficient to do the whole batch of dough at 1 time).
Take a cocktail sausage and put it at 1 end of an oblong at a slight diagonal and then roll up, pressing on the infinitely compliant dough to squeeze it shut, and then place on a nonstick baking sheet, or 1 lined with parchment. Carry on until you've finished all your strips and then get to work with the remaining dough. Three baking sheets should do it.
Now, dip a pastry brush into the beaten egg mixture and paint on the pastry for a golden glaze. Put in the oven and cook for 12 to 15 minutes, by which time they should be puffy and burnished. Remove from the oven and let cool a little before giving them to the children.