No, this is not puff pastry at all. It is a rich yeast dough, similar to the dough for an unsweetened brioche. The dough has a low hydration (around 50% or maybe 55%), so it can be rolled out into a sheet, similar to a pizza base, without being sticky. In Europe, these are made with AP flour, not bread flour, they are supposed to be soft and not chewy. If you are looking for a recipe, search for brioche recipes with a low amount of sugar which are intended for kneading by hand instead of a mixer.
From the rolled dough, you have to cut a stripe, maybe 1 cm wide. You also need molds for baking. They look like a long cone, maybe 8-10 cm in height and 2-3 cm in base diameter. (You can roll your own from tin halfcircles, if you have the tools to cut a tin sheet into halfcircles). You wind the strip of dough around the cone and bake it with the cone still inside. Afterwards, you have to remove the cone with care, and so you have the hole where you can put the sausage.
I haven't made or seen the variation with eyes. Obviously, you have to add the pips before baking. I am not sure how the white/dark blob is achieved. It looks like something was piped on before baking, maybe normal and colored eggwhite "snow".
The shiny effect of the crust of this picture seems to be due to baking with steam in a commercial oven. You can't replicate it at home, if you want them to be shiny, you may consider using a glaze or wash.
I would start by rolling the pastry dough thinner; I often have my dough too thick the first time or two that I work with a recipe. Keep the dough cold to make it easier to work with.
Next, I would try different oven times and temperatures. Try a temperature 25-50 degrees (F) higher, and pull the rolls out 30 seconds to one minute sooner. The nice thing about playing with oven times and temps is that you can go through many tests quickly by just putting a few of the items in for each time/temp combination. (Extra mass in the oven, such as a pizza stone or even some bricks, will help keep the oven temp more steady.)
If the outside of your rolls are getting over done before the inside is done, try chilling the rolls in the refrigerator for 30 minutes or so before baking.
Best Answer
At work the butcher makes 100's of sausage rolls in batches. We then cook 1/4, fridge 1/4 and freeze 1/2. We never have any issue with them being soggy, regardless of storage method. To be honest in a taste test you'd have no idea which was which.
Also once cooked (as long as cooked properly) they don't go soggy for at least 2 days in the fridge.
I'm assuming you will be using puff pastry as per tradition. If so don't forget to allow the rolls to stand at room temp for 15-30min before cooking as it will help with the "puffing".