Question
I have a Pegasus rose that is about 4 years old.  This year, several of the rosebuds have opened only to reveal what looks like more rosebuds growing inside.  I've never seen this before.  (I live on the Olympic Peninsula in Washington State.  It's been cool and wet so far this year.)

Any suggestions as to the cause and/or cure?

Answer
What you are describing is a condition called Proliferation and it is thought to be caused by a hormone gone wrong. The PNW has not had a decent spring but has had a long time with dull weather and no heat. These conditions sometimes cause roses to grow another flower in the middle of the main one. Mostly it occurs on the double roses especially the very old ones such as Gallicas, Mosses and Centifolias but some modern ones are also prone to it.