Intel released the Xeon D series to fill this niche
https://ark.intel.com/content/www/us/en/ark/products/series/...The D-1529 can be TDP limited to 20W (at a whopping 1.3GHz), but if you goal is idle power consumption under 20w you are better off with one of the 35w TDP D-1602 since it is only a dual core and has the lowest standby consumption. The D-1518 is also a popular choice, and basically the same as the D-1529 but with a 35W tdp limit so it can turbo to 2.2GHz.
You can get a barebones motherboard (even mini-ITX is available for making small machines) for about $500 or a complete computer for a bit under $1000 ex https://www.newegg.com/supermicro-sys-e300-8d-intel-xeon-pro...
note - the xeon-d series is not socketed, so the motherboard price includes the CPU. Make sure to get board that takes full size DIMMS since there is not a lot of ECC laptop memory on the market.