The first cut of grass in the spring will have the most Nitrogen. If you have any browns, now is the right time to make a nice, big compost pile that will instantly heat up and start becoming compost.

Make a compost bin using 4' pallets tied into a open-top box, using a doubled strand of baling wire, or rebar tie-wire, to tie the pallets together. Try to fill the bin at least half-way the first time you add material. Chipped/shredded material breaks down the fastest.

Cover the pile with an old leaf bag to help maintain the right moisture content. Don't seal the pile, just put a piece of plastic on the top to keep rain out, heat in, and probably ammonia in.

The ammonia is a precursor to protein, which is required by the bacteria that break down the organic source material. I say 'source material' because finished compost is also composed of the remains of the bacteria that lived and died in your compost pile, along with the bacteria's 'output', long-chain organic acids such as humic (humus) acid and fulvic acids.
Nitrates are volatile, meaning they will evaporate. Too much ventilation of a compost pile will cool it too much, and evaporate the ammonia from the pile. I line the insides of my pallet compost bin with more leaf bags, leaving the corners of the bin open for limited ventilation.

What I've just described was developed by trial and error over an 8-year period, living on a fertile acre of clay soil, in Boise, Idaho. Note that Boise is rather dry and hot in the summertime. Moisture control was important.

I would screen the compost as it reached the end of the active composting phase. Screening helped break apart any chunks of compost, allowing more oxygen, and exposing it to the ambient moisture of the rest of the compost pile. The screened compost would sit for a few months to 'finish'.

If I have some nearly-finished compost in the fall, it is safe to spread about an inch on the surface of soil of a fallow bed. Doing so will help protect the soil from direct exposure to the wind, rain and sun (weathering), and will help incorporate organic material into the soil. Direct contact with surface soil seems to really speed up the decomposition of compost, but in limited amounts (or, at a limited rate). Tilling compost deep into the soil slows down the rate of decomposition because of a lack of oxygen.

Repeat the application of compost to the soil surface over the winter. Don't plant at the same time you apply compost. Give the compost a chance to incorporate with the soil and allow the first, more intense bloom of soil biological activity to die down before planting. During break-down, compost creates short-chain organic acids, like acetic and butyric acid, that inhibit root growth and seed germination.

Eventually, these short-chain organic acids become incorporated into humic and fulvic acids. This helps create the organic slime that denatures clay into crumbly soil that is made of tiny, brown balls of dirt with lots of air space between the balls (peds) of soil. The acids will bind to the types of minerals that the roots of plant are seeking, especially the trace elements, making the mineral nutrients biologically available to the plant. In chemical terms, these acids are helping to 'buffer' the chemical reactions, enabling the chemistry of biology.

Look! I wrote a book!

Good luck with your composting. It's fun when you get it right.