There are numerous routes up Mt Moosilauke & nothing has emerged as a particular "FKT route".
Ben Nephew posted a double traverse option (shown on map):
Mount Moosilauke has a number of trails to the summit, and I choose the Glencliff and Beaver Brook trails for an out and back run over the summit. Glencliff has the greatest amount of vertical climb, and Beaver Brook is special. The posted mileage is about 15.5 miles with 6400' of vertical. The vast majority of the trail mileage is technical. Glencliff is typical NE terrain, but Beaver Brook has a very long and steep technical section which can be clearly seen on the GPS track. It is also incredibly scenic, with the challenging section running alongside Beaver Brook Falls. The views from the barren summit are outstanding on a clear day. I started at the Glencliff trailhead, and turned around at route 112. My time was 3:16:23.
A good resource for Moosilauke and other NE peaks:
From the link above:
Gorge Brook trail, starting at the Ravine Lodge, is probably the most popular trail on Mount Moosilauke (rt: 7.4 miles, 2,550 feet, 5:00). Since Ravine Lodge is the highest trailhead on the mountain, this is the trail with the least elevation gain. It has been relocated recently to avoid the steepest parts of the old trail, and is now relatively easy.
The Beaver Brook trail, though equally short (rt: 7.6 miles, 3,100 feet, 5:20), has an extremely steep and rough section along the beautiful cascades. This section can be slippery when wet or icy.
The Glencliff trail joins the Carriage Road at the base of the South Peak (which has attractive views) and uses the last 0.9 mile of that trail to reach the summit (rt: 7.8 miles, 3,300 feet, 5:30), it has good footing and only one steep section.
The Moosilauke Carriage Road is wide and has good footing most of the way, it also has the gentlest grades of any trail on this mountain (rt: 10.2 miles, 3100 feet, 6:40).
Finally the Benton Trail (rt: 7.2 miles, 3,100 feet, 5:10) has moderate grades and good footing, but a stream crossing near the start may be difficult at high water.
With all these trails a variety of loops are possible. With one car there are two possible loops starting and ending at the Ravine lodge. Go up Gorge Brook trail, and descend by either the Carriage Road and Snapper trail (lp: 7.5 miles, 2,450 feet, 5:00) or by the Beaver Brook and Asquam Ridge trails (lp: 9.6 miles, 2,450 feet, 6:00). A long loop recommended in the White Mountain Guide uses the Benton, Glencliff and Tunnel Brook trails (lp: 13.3 miles, 3,100 feet, 8:10). With two cars it is possible to go up by the very steep Beaver Brook trail and return by either the Gorge Brook (lp: 7.5 miles, 3,100 feet, 5:20) or Glencliff (lp: 7.7 miles, 3,100 feet, 5;25).