From Mihalek's question what I hear is that he's found that ascending 1:1 on a doubled line is more efficient; about twice as fast and easier because you have half the number of footlocks as with a traditional 2:1 system to get up there, and many find footlocking two lines with your feet easier than footlocking a single line. He's found the advantage there pretty obvious. This is DdRT vs DbRT.
Once up in the tree, you flipline in and pull the end of your rope up and fashion a traditional friction hitch which, depending on how high up you are can cost you quite a bit of time in getting the rope up there and configuring your hitch.
The question (in my own words) asks, is there a way to safely attach to the doubled line as it stands in front of you and move on to treecare working off these parallel lines.
The answer is a definite yes. The method works identically to SRT; you just treat the two lines as if they were one. Both ends are on the ground and both lines are not moving, relative to the tie-in point. It is a 1:1 abseil where 100% of your friction is created and controlled in front of you and eliminates the need for a friction saver as there is only the weight of you, your gear and your rope on the crotch. The rope presses down on the tree, but does not slide over the area.
There's a number of advantages in speed, efficiency, minimizing gear and lessened rope wear in approaching ascent / descent in this manner. Ya just gotta figure out how to do it.
Since the thread is about using a friction hitch on this 1:1 system, I'll step out. I might use a friction hitch to back up the friction controller, but I wouldn't use the friction hitch AS the friction controller. This doesn't mean it's not possible, but the alternatives to a friction hitch are easy and inexpensive. The hardest part is the shift in mindset.