Maybe something like
http://asrock.com/mb/Intel/J3455-ITX/ (random hit with a J3455 core, $74 at newegg at the moment)
The base processor is rated at 10W for 4 cores, add ~8W for the 10GbaseT board (plus a bit) and its a competitive solution, particularly if your workload needs more RAM/etc. Feeling like you want a little more beef the Denverton's are hitting the market and many of them have the 10G integrated.
https://www.supermicro.com/products/motherboard/atom/A2SDi-H...
That one also has a BMC, which given my past experience tends to add a few watts too. Without testing them side by side its hard to know which draws more power in a given workload, particularly since the intel machines have become very dynamic over the last couple generations, its actually pretty hard to hit their TDP numbers in most cases (particularly without heavy FP workloads).
Now its become more about what the motherboard manufacture has integrated. Its all fine that the core/etc draws 8 watts. The problem happens when the motherboard manufacture decides to glue an aspeed BMC and an old marvell SATA controller on the board. Between them at idle they draw 2x more power than the SOC does running at peak. Its pretty easy to move the numbers one way or the other with unrelated changes.
EDIT: Discovered after posting that the PCIe on the asrock, is only x1 which keeps it from taking one of the x540T boards, but they have a couple variations including one for less money in m-atx which has a potentially better slot layout, and full size DIMM's.