I am trying to identify / describe this wool thread / yarn
Weaving yarns are described by two numbers separated by a slash, eg 2/5, 3/10, etc.

The smaller number is the number of plies
The larger number represents the size of each ply.

So, a 2/5 yarn has 2 strands of size 5 thread. A 3/10 yarn has 3 strands of size 10 thread. Different sources are inconsistent about which number comes first. Usually the smaller number is the number of plies and the larger number is the size number, but for very fat yarns the size number can be larger than the number of plies.
Each fiber type has a different scale for the size numbers, so you have to look at a table of wool yarn sizes. This table from woolery.com has the smallest wool weaving yarn sizes I could find listed:
Material | Weight | yds./lb. | m/kg   | SETT 
Wool     | 20/2   | 5,600    | 11,289 | 20-30 
Wool     | 18/2   | 5,040    | 10,160 | 20-30 
Wool     | 8/2    | 2,240    | 4,516  | 12-16 
Wool     | 8/3    | 1,490    | 3,004  | 10-15 
Wool     | 6/2    | 1,600    | 3,225  | 8-16 

As you can see, your desired sett of 44 EPI is tighter than the smallest yarn on this chart. I couldn't find a chart that goes any smaller than this, but I know that wool yarns smaller than that do exist. So you know you want a 2-ply wool yarn smaller than 20/2 or 2/20.
When you purchase weaving yarn, sometimes they don't tell you the yarn size. Instead, all they tell you is yards per pound. So it would be useful to know how many yards per pound your desired yarn will have. It will have about twice as many yards per pound as a wool yarn with a sett of 22 EPI. So, probably around 10,000 yards per pound.
The yarn you want is much smaller than what is usually sold for home weavers. Yarns this small are more common in commercial weaving. So, you may want to shop for mill ends, which are yarns leftover from commercial weaving. Suppliers that sell mill ends may also have regular product lines that meet your needs.
Typically with mill ends, once a yarn is sold out it's gone for good. So be sure to buy enough for your project. Here are a few sources of mill end weaving yarns. Some of these I already knew about, but I found others by googling &quot;weaving yarn mill ends.&quot; You might also have success with terms like &quot;remnants&quot; and &quot;production excess.&quot;

Colourmart.com - At the moment they have a 2/42 wool yarn with 10,320 yards per pound. (They don't provide yard per pound measurements, but they do tell you how many ounces and yards each cone of yarn is, so you can calculate yards per pound yourself.)
Oldmillyarn.com has a more limited selection, but they do show the yards per pound right on the search page. At the moment they have some in the 12,000+ YPP range, and some in the 7,000-8000 range.
WEBS is usually a good source of mill end yarns, but they don't have any at the moment. Probably something to do with COVID and supply chains. In their regular weaving yarn section they have a 40/2 linen, but no wool yarn even close to as small as you want.

Note: If you are making a garment or something you want to be soft enough to touch skin, be sure the wool you purchase is garment-quality wool, not rug-quality. If the product description doesn't specify, contact the seller before buying. Rug-quality wool (often just called rug wool or carpet wool) is very scratchy.
If you have a hard time finding wool yarn for your purpose, consider looking at other fiber types. The narrower sizes of yarn are more commonly available in silk, linen, tencel, cotton and rayon than in wool.