If you've baked your puff pastry in the past at 375 then it's probably not a temperature problem, however I usually bake it between 400-425 degrees (400 convection or 425 in standard oven). Lower temperatures aren't a problem for the baking aspect as it will still cook through. The problem lies in the fact that the heat won't be significant enough to generate the steam needed for expansion.
Egg wash could be a culprit as it should only be used on the top and not allowed to run down the sides where it will seal the edge together and keep it from rising unevenly.
I'm going to take a wild guess at what's happening, based on what I know of thermodynamics, (but I could be wrong about the whole thing)
When a cakes and bread bakes, it bakes from the outside ... as it forms a crust, it can no longer rise. (for yeast breads, this is why we slash the top, so it can get a little extra rise out of it)
With a cake, as the sides set, it'll continue to rise in the middle, creating a domed top. With a metal tube pan, the heat from the outside of the pan conducts to the middle of the pan, so the middle will set shortly after the outside.
...
With the silicone pan, you've actually got an insulator ... the heat's not going to conduct to the inside, so it's going to be able to rise longer than around the inside -- basically, the domed cake, but you've got something for it to stick to, so it doesn't fall back down as it sets.
As I don't think the issue is distribution of the batter, I'd recommend trying other techniques to reduce the domedness of the cake -- cook it at a lower temperature (try 25F / 15C lower) for a longer time. You can't remove the problem entirely, but you might be able to reduce it.
I'd also look at the instructions that might've come with the pan -- it's possible that they have reommendations on how to deal with the problem. (and it's possible that this might not be as significant a problem in convection ovens, but replacing your whole oven for a set of pans is kinda silly)
Best Answer
"Pastry bars metal" seems to work. The results I got were for "caramel ruler bars" or "ganache ruler bars". Other terms are "confectionery rulers" which gave me a forum that recommended getting them cut for you at a metal shop from stainless steel at whatever lengths you want for less than buying purpose made ones.
That said, I'm not sure that's actually what you're looking for. These seem to be bars for rolling out dough or fondant to a specific thickness, not designed to be used for shaping things the way your photo implies.
The term that I found that more closely resembles what you're looking for is "confectionery frames".