I like the urine and molasses-water ideas.
Just learned that the thermophilic bacteria (the ones that initially heat things up hot enough to kill weed seed) love sugars.
The next round likes the complex carbs and protein and then after it's cooled down you get the fungi working on the cellulose, lignin, etc (the tough woody bits).
Fresh grass (that is still giving off that wheatgrass smell) should have sugars still present.
Flowers do too - if you are trimming
dandelions out of the lawn, or pruning, or have any flowering weeds, those might help if they go in fresh.
You could also add apple-peels, cooked potato-peels, etc, if you can get them.
Let's see.... what makes tough bits into sugars faster? Mushrooms do sound good. If there's not enough of the N and sugars to heat it up, maybe you can even skip ahead and get the mycelia going now. Then once the sawdust has broken down a bit you might see the pile revert to what you'd expect from compost, bacterial heating-up.
Spit has an enzyme that makes starches into sugars, called amylase. Got any herbivorous animals that drool a lot? Maybe you could spit in the
bucket of molasses-water a few times, let it ferment a bit, then dose the pile.
(I am totally making this spitting-in-the-molasses part up from unrelated knowledge-bits, but I think it could work. Couldn't hurt, anyway.)
Presumably if you had a lot more kitchen scraps, you'd be using them. I've heard of using molasses and cornmeal, or any grain powder, under black plastic as a super-hot composting weed-suppression method. Worked pretty well when our landlady tried it in Portland, and you can get horse-feed grade molasses that's very reasonable by the 5-gal bucket. Any grain - maybe flour that got meal-moths in it, or whatever - will
boost those sugars, and if applied by rinsing it through the layers with water, it should kick-start pretty soon.
I think the moisture will help more than anything.
Our dry climate means we routinely have to wait several years; Mariah Cornwoman puts sprinklers and tarps on her piles, layered for aeration, and gives them about 3-4 years. The dry parts stall out, have to be rearranged and sprinkled again. (artesian well, not city water)
Note on experience level: I am not a big-time composter by any means.
Since I rarely keep up with the whole garden area anyway, I've started just piling things up directly over one of the sunken log/dirt/mulch beds, in hopes of having a kick-ass mulch bed by the time I work my way over to that side of the plot some years hence. I sheet-mulch with the weeds and old
hay, and sawdust / wood chip as well as old hay goes around the blueberries. Acid-loving little pets, they are.
My latest big pile of kitchen scraps (we had a freezer go unplugged last month) just went into the new
hugel bed.
I know the theory of composting to kill weed seeds etc, but with our very cold winters I have about half the year to work with; and with the dry, dry summer and fall it's realistically about 2 months (late April - early June) that compost would naturally happen around here. The rest of the time it would need sprinklers, then a big insulated pit like a reverse ice-house or something. And I'd need to move it at least twice, not to mention turning it.
I find that seed-grown weeds are not that hard to pull, and any bare dirt gets seeded by all the other weeds in the pasture anyway. So I don't see much point in putting the time into a weed-seed-free compost, when I can mulch with the same materials and see the dirt improve in a quarter of the time. The mulched beds have delicious soil, and the current crop of weeds just became more mulch, with today's seeds planted where the weeds got pulled.
This is a small-time hobby garden; I could totally see it being more interesting at the industrial scale, if you're farming with more equipment and less labor per acre.
-Erica