In a small sauce pan, cook 1 cup water + 1/3 cup rice for about 10 minutes. Drain excess water and set aside.
Cut off the top portion of the pattypan (where the stem is), just about 1? thick.
Using a small knife create an outline of the opening in the pattypan around the “rim” at the top, leaving about 1/2? margins. Using a regular tablespoon, scoop the pattypan’s flesh out along the margins and hollow it out throughout. You will have to reach further into the pattypan to scoop out the sides, but it doesn’t require a huge effort — pattypan’s flesh is quite tender. Reserve the scooped flesh, and sprinkle some salt inside the hollowed out pattypans, and rub it into the sides and the bottom.
Cut the meat into small cubes and grind in a processor.
Chunk pattypan scooped flesh, onions, garlic and carrots and pulse them in a processor until small crumbs form. The mix will be quite liquid at that point.
Melt some butter or vegetable oil in a skillet over medium heat.
Add ground meat and vegetables all at once. Stir well until combined in a uniform way. Add thyme sprigs, cumin, paprika and salt, stir well, and saute for about 7-8 minutes, until all juices are evaporated and meat is nicely browned, but not too dry. Remove thyme sprigs at this point.
Add rice and chopped herbs to the meat and stir.
Fill the pattypans with stuffing quite tightly. It won’t expand anymore since we precooked the rice.
Cover the stuffed pattypans with the cut off top portions, using them as lids.
Wrap each pattypan in a square sheet of foil, gathering the four corners of the sheet above the top of the pattypan.
Bake stuffed pattypans in a 375F degree oven for about 45-50 minutes, or until pattypan is fork tender. In the last few minutes of baking, open up the foil slightly to allow outside of the pattypans lightly brown. You can use the broiler for this, if desire.
Serve hot, with an [optional] dollop of sour cream or plain yogurt on top.