If (hypothetically) I wanted to use January 1st 12:00 (noon) 2010 as my epoch, how would I calculate this number?
The USNO page refers to USNO Circular 163 which in turn refers to a paper by Aoki et al. and one by Moyer. These papers explain how the expressions created by Simon Newcomb at the end of the 19th century were updated to work with the new system of constants introduced in 1984. If you want to change the epoch, you could redo these calculations with the epoch of your choice.
Further changes to celestial mechanics calculations have occurred since then, but apparently the USNO has decided that for the accuracy stated in the USNO web page, 0.1 second, it is not necessary to overhaul the calculation to account for all the changes since 1984.