Stuff like bay does rot down, but it is never quite as fine as compost made with softer green waste.
I have quite a bit of bay in my garden at home and truck that, plus other woody chippings, to the plot to compost. I keep it together in a single bin and add in greener material. Grass clippings would be fine. I cover the bin with split open compost bags weighed down with bricks and it takes at least all winter here, or into mid spring in a cold year. It is warmer where you are, so you might find it quicker to rot down.
I use this compost for mulching areas for growing things like squash or tomatoes, but it would not be ideal for small seeds. By the following year, it has disappeared into the soil and does add a lot of heart to it.
