So it turns out that this is a Tree Peony:

There were decades-old peonies at my old house and I was sad to leave them.  I was happy early this spring when I saw the small purple peony buds peeping up through the wet, cold leaves.  But THEN, I found a tag (left by the previous owner) for a "tree peony" and I've been watching any likely bush or shrub for enormous, fragrant flowers.  Yesterday, I pulled in the driveway and saw the bright purple flowers from the above plant.

I clipped one of the low-hanging flowers and brought it inside.  After a couple hours in the sun on the dining room table and in a bud vase of fresh cold water, this flower opened fully.  I measure it - it is 8 inches wide!  That is one serious flower!