Sounds like you'll only have one mash/lauter tun. If it is not around 15 gallon capacity you'll have to do two separate mashes/sparges. Or, you could get a paint strainer bag and do a BIAB mash in your 10 gallon kettle same time as the mash/sparge in your mashtun, and combine the runoffs into your friend's keggle. Hopefully you have a good chiller, I use a 50' long 1/2" copper immersion chiller in my keggle for 10 gallon batches. I have had to use a 25' for 10 gallons before. If that's what you have, just keep stirring the wort while chilling to reduce the time to get the wort to pitching temp.
If you don't want to do a concurrent BIAB, just do two separate mash/sparges. After the first one, I would keep the runoff in the keggle above 160F so nothing can grow in there until you add the second mash/sparge runoff.
I'm no expert on BIAB but I don't see why this wouldn't work. Or without using a strainer bag, mash 1/2 of your wort volume in the 10-gallon pot, runoff through a strainer into the keggle followed by a sparge and again runoff through the strainer.
After chilling, just transfer from the keggle into two fermenters in equal volumes and pitch as usual with different yeasts.
If you want to brew two separate beers with different grains and hops of course you'll have to mash/sparge, and then boil them separately, each for 5 gallons of wort.