What type of protein specifically? Any type?

You may be sensitive to meat with high levels of tryptophan. Tryptophan is an amino acid. Protein is a combination of different amino acids. Tryptophan is a precursor to the neurotransmitter Serotonin and increased levels of serotonin can make people tired.
Some meats and foods have much higher levels than others.

This list may help. http://wholefoodcatalog.info/nutrient/tryptophan/meats/low/1/

Also, it may be a digestion issue and you may need more stomach acid to digest the protein. Many people don't have adequate levels of stomach acid and various digestive enzymes.
HCL/betaine taken with protein meals can be very helpful.

Quick question first: Does this also happen with carbohydrate? People with poor insulin sensitivity are often made lethargic and sleepy (as opposed to energetic, warm, and happy) after a meal containing carbohydrate.

Too many calories at once? Food drunk.

Does this apply to all proteins or just protein from dairy? Either way, you should take steps to increase your stomach acid production.

I would get very sleepy after a high protein meal too. I didn't get my energy back until I reduced protein and increased carbohydrate. It's funny that protein messes with me like that but carbohydrate doesn't.

Great question, way to trust yourself and not listen to common information!

Anyway, I'm realizing this is why I'm failing on paleo is because it really limits you to eating way too much protein, and after a year, I realized my sleepiness is probably because I eat so much freakin meat.

I'm noticing improvements decreasing the protein, and eating some SAD foods, which cause some pimples, sinuses problems, but it's a tradeoff as it does give more heaviness to my body (a good thing for me) and makes me feel more connected to the rest of the world, as everyone else eats SAD foods, it biologically assimilates you.

Some protein like turkey raises levels of serotonin (I think it's the L-tryptophan in the turkey). Could something like this be happening?

This happens to me after I eat eggs. I just did about 30 min ago and I'm ready for an hr nap! If I want to avoid being tired I just eat a veggie with a different protein. Like adams all natural peanut butter.

The liver turns excess protein into glucose, which could trigger an insulin spike. Most only actually need about 6-7oz of protein a day- no more than 2-3oz at each meal. Anymore than that and you risk spiking glucose levels and consequently, insulin levels, which could result in the sleepiness you described.

Get your liver function checked. If you have a sluggish or fatty liver then it cannot process protein and releases ammonia into the blood. This triggers the brain to cause a feeling of fatigue or drowsiness. It's a symptom common for liver diseases.

You have said you are only eating lamb and fish...does this also happen when you eat beef/chicken/pork etc...? Also, you mentioned you are eating grass fed lamb, but maybe you should try lamb from another source, and also check the source of your salmon/fatty fish. Would you consider yourself fat adapted?

You mention recovery. Protein requires a good bit of energy to process, so I suspect when you eat it the body is already interested in shunting available resources to your recovery, then it has to process the protein. Consciousness also takes energy.
I think this is why a lot of people lose their appetite and sleep a lot when they are sick. Go with it. If you are not hungry, don't eat. If you are hungry, well I'd wait until before bed to eat protein. It will probably go away as you get better.