I took loads of pictures - LOADS of pictures - during construction so here's a kind of tutorial type thing.
I was aiming for something kind of like the center bonnet in this fashion plate (the blue and white one - everything I own is blue and white).
So I cut that brim in half, and laid the halves ugly sides together.
Then I "pinned" the halves together around the outside brim edge with binder clips. I started with one in the middle and then bent the whole thing against my work surface to clip the rest.
So then I got out some thread (in a random teal color b/c who cares not me) and stitched the brim(s) together around the outside edge. I mostly went through the fabric parts and also a little the edge of the cardboard. I mostly avoided the cardboard because cardboard isn't really a party to sew through.
Then it was time for the crown. I needed to shape the back of the brim so the crown attachment would sit in the right place on my head. I forget exactly how I did this; I think I basically just set it on my head and guessed. I drew on a new crown line and cut it with tin snips - I was hiding from the heat in the basement, where the hardcore tools are, and anyway it would have been a nightmare to cut even with sharp paper scissors (of which we have 0).
Not a lot too small, either, just an infuriating couple inches. So I went upstairs into the gross oven world and retrieved a bit of more fabric.
So I cut a new crown like 3 inches bigger than the other one and clipped it to the brim, and sewed it down. Because, as mentioned, cardboard is not a party to sew through, it required some convincing with my awl. But I got the crown sewn from one side to the other and then I went back the other way again (which is sort of cheater's saddle stitch) to squish down the freehand gathering that was necessary to get the pieces attached.