Hi! Congrats on the new AR-10!
I won't claim this is the "right way" but I think the shoot 1 round, clean, 1 round, clean, etc for 20 shots is ridiculous IMHO. I don't mean to offend anyone who uses this procedure, I just think there is little benefit from it, especially in a non-super match, custom barrel, benchrest gun. The factories do a pretty good job on barrels these days, especially on a nice gun like yours. I also think the main break-in affects the action by "wearing" it in a little and helping to reduce copper fouling in the barrel. There will be dozens of ways to do it depending on who you ask, this is just how I did it.
Clean thoroughly before shooting, get all of the preservative/factory oil out of it first (I think this is the most important step that most people overlook).
Dedicated AR cleaning brushes really help, especially the chamber/locking lug combo brush, called 7.62/308 chamber brush. Iosso makes a kit for AR-15s that has 3 brushes for cleaning the bolt and carrier that works decently on the Ar-10, it will include a chamber brush, but that is too small for the 308. A short cleaning rod that doesn't have a swivel handle is your friend for cleaning the upper/chamber. I've tried regular ones for pistols/rifles and just got frustrated. Dewy makes a nice one made of brass, I'm sure there are others. Some people use dental piks for scraping stubborn carbon from small places. Q-Tips are also helpful. A bronze or nylon brush (like a toothbrush with two ends, small and large is mandatory) For carbon removal, I like Pro MP7, it is thick like a gel you spray on, let sit for 15-30 minutes and the carbon comes off pretty easy, plus no smell, no harm to you gun.
Lube bolt, bolt carrier, firing pin, inner upper receiver, and locking lugs. No need to lube the barrel. No need to have lube dripping out of it, just an light even coat on the lugs, buffer spring, and lower parts, a little more in the receiver and bolt assy. I use Break Free CLP, same thing the military uses for M16s both to clean and lube. I've fired 300-500 dirty blanks in an M16 with a BFA without cleaning, just a squirt of break free now and then and it keeps running. There are many great lubes out there, maybe someone will chime in and give you a few others to use.
Shoot 20 rounds to sight in/test function, do NOT overheat barrel (no 20 shots in a row rapid fire), 5 shots slow fire, let cool till just barely warm or cold, 5 more same cooling period, etc.
Clean gun, especially the barrel, get the copper fouling out of barrel, you can usually see it with a bright light at the muzzle after cleaning the carbon out.
Clean barrel normally until carbon fouling is out.
Put the copper remover on a patch and run it back and forth through bore a few times. I use Sweets 7.62 copper remover. Let sit for 10 minutes, no longer than 15. Use a bore brush and scrub barrel (i also like nylon since it will not dissolve in the Sweets, you can use bronze but if you don't clean the brush thoroughly after use it will fall apart, plus the blue color will never go away from the bore if you keep using it after each patching). Run patches through barrel until dry. Repeat until patches no longer have any blue color on them. Run a few more patches with regular bore solvent on them after the copper removal to ensure the copper solvent is out of the barrel.
Clean bolt and locking lugs, lube same as above.
Go shoot 40 rounds using same procedure, don't overheat it. If it feels hot when you grab the barrel, stop and let cool.
Clean same as above procedure.
Do the 40 round procedure again.
Then finally, enjoy your nice AR-10!
This procedure has worked for me, my AR-10 can turn in 1/2", 5 shot groups with several loads. I don't like to shoot more than 100 rounds without cleaning, but I am all about pure accuracy and looking for the elusive 1/4" group lol. I love blasting with my AR-15 carbine, and am a lot looser on my self imposed rules with that, all depends what you are looking to do.
I have also used this procedure on a Remington Sendero and 700 Special R-5 Milspec. The Sendero is a 3/4-1" gun, 300 Rem Ultra Mag (might be better but the recoil dosen't help
The Milspec (308) will shoot under .5 MOA with ease, I am the weak link with that gun.
Hope this helps and gives you some pointers, there are many people here more knowledgeable than I, perhaps they will chime in and give you some more tips and tricks.
Enjoy the new gun and welcome to the forum!