Is there a way to force generated soldiers to always be drawn from my character pool?

This is no longer required, since they added options to the character pool menu in the latest patch. See here.

Yup, it's possible, and really easy to do too.

Open up \path\to\steam\SteamApps\common\XCOM 2\XComGame\Config\DefaultXCOMGameData.ini, and then find the following lines:

InitialSoldiersCharacterPoolSelectionMode=eCPSM_Mixed
RewardUnitCharacterPoolSelectionMode=eCPSM_Mixed
...
RecruitsCharacterPoolSelectionMode=eCPSM_Mixed

Update these values to the following:

InitialSoldiersCharacterPoolSelectionMode=eCPSM_PoolOnly
RewardUnitCharacterPoolSelectionMode=eCPSM_PoolOnly
...
RecruitsCharacterPoolSelectionMode=eCPSM_PoolOnly

Changing all of these values to CPSM_PoolOnly will ensure characters will always be drawn from your character pool if available, but will revert back to random generation if there's no unused soldiers in the character pool (source).

The other answers were correct but now there is an in game setting for this on the character pool screen. This was added in the March 10, 2016 patch.

This is now a Steam workshop mod for this: Character Pool Only