1. Pour the stock/water into a large saucepan or pot and let it warm up a bit before whisking in the cornmeal (I like to add it in a slow stream to try and avoid lumps). Bring to a boil, stirring frequently until polenta thickens. Reduce the heat to medium-low, cover and continue to cook for 45 minutes, giving it a stir every once in a while.
2. Stir in the olive oil, cheese and rosemary and pour into an oiled 8x8 pan. Press a sheet of plastic wrap on the top and let sit in the fridge for at least 4 hours, preferably longer.
3. Cut into squares (size will depend on what kind of buns you're using!) and pan fry for a few minutes on each side until warmed through and crisped up. Serve on buns with arugula, compote and more goat cheese if desired.

To make the caramelized onion and fig compote:

1. Heat a large pan over medium high heat and add onions along with olive oil and 1 tablespoon of butter. Stir everything to combine.
2. When the onions start to break down a bit, add the salt and reduce heat to medium-low. Cook for 20-25 minutes, stirring frequently - you want the onions to slowly turn brown and caramelize without crisping up. If it seems like it's going to fast, lower the heat even more.
3. Add the figs and continue to cook, stirring frequently, for another 20-30 minutes until everything gets really soft and starts to cook together. I like to mash some of the figs during this time with the back of a wooden spoon.
4. When everything is cooked down to your liking, add the vinegar and cook for another 5 minutes.
5. Store leftovers in a tightly-capped jar in the fridge.