Dry stout from dried malt extract and a mash

I originally intended to make a dry stout from dried malt extract, but when I realized I didn't have enough extract, I decided to try doing a mash of what grain I had available to make up the balance of sugars. The results:

Initial calculations of conversion between dried malt extract (DME) to grain amounts

I started from the "Toad Spit Stout" (pp. 207-8) in Charlie Papazian's "The New Complete Joy of Home Brewing.  It calls for these amounts of malt extract:

4 lbs. dark dried malt extract (DME)
3.3 lbs. liquid malt extract

I wanted to just use DME (dried malt extract), so I used a conversion factor of 0.84 to estimate I could use 2.8 lbs. of DME in place of the liquid malt extract, yielding a total of

6.8 lbs. dark DME

I only had 5.8 lbs. of DME.  However, part of the recipe is some grains (1.4 lbs.), so I had the idea that instead of the normal process I could mash those grains to extract their sugars to make up the difference.  How much sugar would I get from this mash?  To estimate, I looked up an all-grain recipe for dry stout (https://www.homebrewersassociation.org/homebrew-recipe/brun-dry-stout/), it calls for 8.675 lbs. of grain, and assuming it would use about the same amount of sugar, I can use this equation to calculate the rough equivalent between DME and mashed grains:

(8.675 lbs. grain / 5.5 gallons) * (5 gallons / 6.8 lbs. DME)
= 1.2 lbs. grain / lbs. DME

I need the equivalent of 1 lbs. of DME, therefore I should use 1.2 lbs.  I had 1.4, so I'm calling that close enough - given variability of me running a mash for the first time, I just wanted to make sure I was close enough.

Mashing

The rough rule of thumb I wanted to use, based on Papazian's notes (p. 254) is at 70 C mash for 15 to 25 minutes, at 65 C mash for 45 to 90 minutes.  Since I was going to keep 70 C as my upper limit, I aimed for 45 minute mash time.

I cracked the grains using grain mill attachment for a Kitchen Aid, set on the coarsest setting, with the Kitchen Aid running at the slowest setting (like this).  I then put them in a pot in ~1 gallon of water, and heated to 70 C, and then turned off the heat.  I measured the temperature approximately every 5 minutes, stirring to try make the temperature uniform. 

At 24.5 minutes the temperature had dropped to 67 C, so I turned the heat back on (2 out of 6 on the dial, so medium low).  at 29.5 minutes the temperature was 68.5 C (68 C at the top of the pot, 69 C at the bottom), turned the heat down to 1.  At 37 minutes, the temperature was 71 C top, 72 C bottom, so I turned the heat off and started a simple sparge.

For the sparge I strained out the grains, then rinsed the grains with cold water twice - with the grains in the pot, I filled with water to just cover the grains, stirred vigorously for 2 minutes, then strained the water out of the grains and mixed with original mash water.

Wert(ing)

The amount of liquid I had now was too much for my pot, so I transferred ~4 cups into a smaller side pot to boil it.  The remainder I put on medium heat in the original pot, I added the 5.8 lbs. of dark DME and 2 oz. of Perle boiling hops.  After a ~5 minutes from when boiling started a foam formed and threaten to boil over so I turned the heat off.  I stirred, and after it subsided turned the heat on medium low.  At the 30 minute mark, there was enough space in the main pot that I was able to pour in the wert in the separate pot.  I kept stirring as foam would form on the surface, and at the 50 minute mark added the 1 oz. Fuggle hops.  Turned off the heat at 60 minutes.

I then poured the wert into a carboy that I had 2.5 gallons of cold water.  I then topped the carboy up with cold water (to get ~5 gal. total liquid).  After stirring vigorously the temperature was ~94 F, so I covered the carboy opening in foil and waited ~45 minutes for it to cool to 84 F, at which point I poured in the yeast and stirred vigorously.  I then set the carboy up with an overflow airlock (stopper in carboy attached to a tygon tube whose other end is submerged in a pitcher of water).  The next morning it was bubbling merrily!

Ingredient list (when all was said and done)

5.8 lbs. dark DME

0.75 lbs. crystal malt (110 degrees)

0.33 lbs. roasted barley

0.33 lbs. (another dark roasted grain can't remember what exactly)

2 oz. Perle hops (boiling)

1 oz. Fuggle hops (finishing)

1 packet ale yeast