So since we can't really get away from chemistry, let's look at your pile. :-)
Composting needs carbon, nitrogen, kitchen scraps, manure (if you have it) water and soil.
You've got egg cartons (carbon) and newspaper (carbon), and nitrogen breaks down carbon. Wet carbon breaks down faster than dry carbon, so if we find a wet source of nitrogen -- some organic fertilizer from a plant nursery, like a box of Dr. Earth plant fertilizer dissolved in water -- then soaking that carbon in wet nitrogen, putting it over the kitchen scraps. Or a bag of composted cow manure. And over the top of all of that put your really great clay soil.
Lasagna layers. If you search on Lasagna gardening you'll find lots of great layers of soil amenders and composting ingredients that will work.
Compost wants to be damp, but not wet. You've also got really great mineral-filled clay soil, so don't despair! A good three inches of that over the top of your pile will keep the flies to a minimum. If you are patient enough, you'll notice birds will start hanging around those flies and eat them!
If a pile dries out, it won't compost, so be sure the ingredients are covered and damp.
It's not a fast process, so be patient.
Mediterranean climate, hugel trenches, fabulous clay soil high in nutrients, self-watering containers with hugel layers, keyhole composting with low hugel raised beds, thick Back to Eden Wood chips mulch (distinguished from Bark chips), using as many native plants as possible....all drought tolerant.