[PureHarvest]

Nathan, there are big differences in poultry waste products

I deal with poultry producers often in my line of work. We write nutrient management plans for them along with other planning stuff.

Your info is somewhat incomplete to make some decisions.

Hatcheries here only hatch the eggs, there are no layers on site. Their waste is dead chicks, egg shells, and feathers.

A egg layer operation whether for eating or hatching will be basically the same for broiler operations (meat birds).
Birds are in confinement, poop on a floor that has some level of wood shavings. The house is either crusted out or wind-rowed after each flock, and the resulting product is manure and litter (wood shavings). Due to not wanting to replace shaving often, most of the load is actual manure. Here, this product is typically put into a manure storage building that is roofed but open on one end. The pile can then decompose a little.
Typically, not a lot of decomposition occurs because the c:n ratio is not correct, because the load is high in N from manure and low in C from not a lot of wood.
This product is then spread onto farm fields. Some do it in fall, some in spring. A rough average here would be 2-3 ton per acre, so there is little chance for burning.
Ammonia volatilizes extremely fast if the manure is not worked into the top 1-2". You can lose 50% in a day.
Some states require manure be worked in within 24 hrs so this loss does not occur.
Layer manure will have much more calcium than broiler litter. This might be good or bad depending on your soil and how much you use per given area its spread on.
The take away is: is your pile wet (layer littler is sloppier, where broiler is dry)?
Is their a local lab (like here) that can test a sample (like $12 here)?
Do you have a test for your soil?
What is the sgft of your garden.

Without knowing some of that, you are kinda flying blind.

If I had broiler litter, I'd use 50 lbs per 1,000 sqft. This would give you 100 lbs of actual N per acre (broiler litter here is about 4% N). Only 50% of the N is available in the 1st year, so really you're getting 50 lbs.
Tomatoes need around 100-150 lbs of N, depending on soil type.
If you worked that into the top few inches, you could safely plant this season. The only reason you would not is because of bacteria. Below ground crops need 120 days (if i recall, might be 90) for bacteria to be killed by soil fauna for the produce to be safe. Above ground crops I think is 30 or 60, i dont recall. So for maters, you'd be safe due to the fruit being above ground typically after 60 days.
If the