How to make a shaped felt hat:

The Hat Form:- Find a ball the circumference of the doll's head (e.g.tennis ball, toy ball, foam ball). Cut it in half and glue or tape it to a flat surface like a dinner plate, a pizza pan or just a board. If you can't find a ball the proper size, just build a half sphere with wads of newspaper, taped to shape with masking tape. Cover the form with a bit of Saran Wrap so that the felt form won't stick to the form when it is stiffened later.

Thoroughly wet a circle of felt in very hot water, squeeze it out and tug, pull and smooth it over the form.  Let it dry and stiffen it by slathering fabric stiffener or white glue (like Elmer's) all over it. Work it into the felt and wipe off the excess. Let it dry. 

The Brim: cut a larger circle of felt with a hole that is slightly smaller than the circumference of the crown. Wet it hot water, squeeze out the excess, tug it over the crown to lay flat on the base. Let it dry and stiffen it up. Let dry and cover the join with a pretty piece of ribbon.

Addendum: 1) Wet felt will stretch over almost any uncomplicated shape (round, oval or flat top). 2) Another way to make a slightly wavy brim is to make the crown circle much larger, put rubber bands at the base of the crown and tug the brim out from the crown fabric.
  
Personal Note - This project doesn't take any time at all and requires no sewing. There is another way to stiffen felt but I haven't tried it on a doll hat yet.....Years ago when loggers still wore black felt fedoras instead of safety helmets, my good fella waterproofed and stiffened his every year with candle wax: He'd slather melted candle wax all over the hat...stuff it with newspaper and put it into a low oven for 10-15 minutes to run off the excess wax...It worked great!