Looking for companies that offer home construction loans, where
interest on the construction payments is rolled into the end loan. 
Specifically, I make no payments (other than the down) until the house
is completely constructed

Maybe you can get a separate short term loan from the bank during the
construction, and this plus interest will be settled by payout of the
mortgage loan.

Don't know if you are in the US, but if you own your own land, there
are several builders who will do that for you.

One, nationwide, is Jim Walter Homes,  google it for their web address.