What is a protein rest?

Let's take the easy bit first - the sparge makes a mashout redundant, since the liquor will be above 170F. On a homebrew scale, dedicated mashouts are often of no benefit, and quite a bit of hassle if you have to do another infusion.

I'm surprised the recipe calls for a protein rest. A beta-glucan rest at 110F is probably more appropriate, since that stops the oats from becoming gummy, improving lautering and can improve efficiency a few points with the unmalted grains.

You have to decide yourself if the infusion worth the hassle. Many brewers these days stick to single infusion and use rice hulls to help with lautering. It makes for a smoother brewday.

If you choose to do a double infusion, for calculating infusion sizes, you can use brewing software, such as Beersmith, or this online infusion calculator.

Now, to answer the question - what is a protein rest?

The grain contains chains of amino acids which form proteins normally used by the germinating seed. In brewing, these proteins on their own and in forming complexes with hop acids, contribute to head retention in the beer. The protein rest activates enzymes that break down the proteins into amino acids. A little is good for the yeast, but removing too many proteins negatively affects the head retention of the beer. A beta-glucan rest is more appropriate with unmalted grains, since it allows the starch to enter solution more easily and removes the gumminess.

See

• The protein rest and modification, How to Brew, J.Palmer

mdma's answer is good, you may not actually want to do multiple infusions or mash out, but I don't feel it solves the implied question of "why would I want to do a protein rest? For this, I'll simply quote from the link he provided to Palmer's site:

The typical Protein Rest at 120 - 130°F is used to break up proteins which might otherwise cause chill haze and can improve the head retention. This rest should only be used when using moderately-modified malts, or when using fully modified malts with a large proportion (>25%) of unmalted grain, e.g. flaked barley, wheat, rye, or oatmeal. Using this rest in a mash consisting mainly of fully modified malts would break up the proteins responsible for body and head retention and result in a thin, watery beer. The standard time for a protein rest is 20 - 30 minutes.

For a deeper understanding of malt modification, I recommend picking up the current (3rd) edition of Palmer's book. It's worth reading cover to cover.

As for raising the temperature, yes, you would add boiling water into your cooler tun. It will thin out the mash, so plan on a low grain-to-water ratio for the initial infusion and a higher ratio for the sacch rest. Yes, the saccharification rest is the same as the main mash that most homebrewers use. It breaks down the sugars (saccharides) in the grain.

Use software like mdma recommended to figure out the volume of boiling water needed for infusion volumes. Palmer's book has the math worked out the long way; I did the manual calculations on my last brew just for the heck of it and missed by nearly 10 degrees, but had extra boiling water handy to make up for it at the cost of a very thin mash (~2.25 q/lb). I think my problem wasn't the calculations as much as more heat loss from my tun than I expected.