I ate a scotch egg at thebreslin.com in New York City.
The dish was clearly cooked fresh and the yolk of the egg was still runny when I cut into it. It was pretty amazing.
My question - how can I get the yolk to stay runny? Scotch eggs are pretty easy to make with hard-boiled eggs - but soft-boiled? Is it just a timing thing?
Timing and precision are key. First off, you need to boil your eggs for exactly 5 minutes, assuming they're large. They should be at room temperature before you start, and you should let them cool afterwards. This should result in a cooked white and a very runny yolk before you fry.
The oil you use to fry the Scotch eggs needs to be just the right temperature - too hot and the crust will brown before the sausage is cooked, too cool and the yolk will cook solid before the crust is browned. The oil needs to be 350F/180C; a cube of bread should take 1 minute to completely brown.
If the temperature is right the Scotch egg should take about 5 minutes to brown evenly, and you should have properly-cooked sausage and a yolk that's still runny.
The scotch egg with a runny yolk -- or other cooked eggs with yolks of various textures -- are a hallmark of some of the new molecular gastronomy chefs. Be warned, though, that it's probably more effort than you want to go to.
Here's a couple of links to online recipes:
If you can find it, the first issue of Lucky Peach includes a whole long article on egg cookery which explains the various cooking times and techniques.
Everything you try, will require trial and error. Basically practice makes perfect, but here is a method I use:
Freeze the raw egg in its shell, de-shell the frozen raw egg and wrap the sausage meat around it whilst it is still frozen. This can be difficult as the egg's albumen defrosts quickly. Coat in flour, egg and breadcrumb. Fry till the scotch egg floats and the crumb is a deep golden brown.
It's the only way I know that gives a soft runny yolk every time.
(This is more or less an addition to ElendilTheTall's answer, but it's too long to fit into a comment.)
My experience with eggs is that "let them cool afterwards" won't fully do the trick, since the egg yolk will simply continue to cook from the heat left in the egg. To prevent that, after boiling for five minutes, you need to cool them down quickly by putting them under running cold water. This stops the cooking process, and when deep-frying later on, they will just heat through, but the yolk will stay runny and not start to set anymore.
See also here in section "Cooling".
For a recipe, see here for an example. When this was broadcast (last month I think), there was a lot more emphasis in the programme on how important it was to cool the eggs than the sentence Drain and cool the eggs under cold running water suggests.
The easiest way to create a runny center is not to use a boiled egg, but a poached. Use cling film and a small bowl, place your egg and wrap it in the cling film and tie it off. Cook your poached egg for 2 minutes. let it cool and snip away the cling film. place the egg in the freezer and allow it to set. The egg is only half cooked at this stage. Wrap your egg in sausage meat and breadcrumbs and deep fry for around 4 minutes. You will have a perfect runny center and a treat!
I make them by the dozen myself. I gently put the egg into boiling water for 4 minutes then straight to an ice bath for about 10 minutes. once chilled gently crack the shell all the way around the center and put back into ice water or 5-10 minutes then peel. Wrap in thinly pressed sausage and directly into breading and refrigerate. I cook at 300 degrees for 5 and a half minutes. No complaints so far. Not really too hard or too time consuming once you get the hang of it.
