Unfortunately for you, there is no such opening.
You see, the problem with
1.d4 is that
d pawn is protected from the very start, unlike his "colleague"
e pawn. While it is possible to cut down on theory learning against
1.e4 by simply attacking the
e pawn ( Alekhine's defense, Scandinavian defense, Petroff defense in a way ) thus forcing White's response, you can not use the same method against
1.d4 since the pawn is protected as I mentioned earlier.
So your only chance is to find an opening that has the least amount of theory to learn, but be warned--all good defenses against
1.d4 have lots of theory worked out to move 30 and sometimes beyond. Just because the number of branches is small does not mean that the sub-variations are short.
Yet another problem with
1.d4 and an important one--I use this all the time to quickly demolish opponents with weak opening knowledge--most of the good defenses to
1.d4 can be dodged by careful move order transposition!
Your problem is far more complex because of this. Currently
Nimzo-Indian scores very well against
1.d4 but can be dodged if White plays
3.Nf3 instead of
3.Nc3. Then you are forced to adopt
Queen's gambit declined or
Queen's Indian defense and both of those openings give slight advantage to White, which severely reduces your chances to win.
King's Indian defense still holds but is prone to become "refuted" from time to time, only to be revived again with some great novelty. Still this defense, along with
Queen's gambit declined does offer you one benefit other defenses do not:
White can not use transpositional tricks to kick you out of your opening!
You can play both openings against anything White throws at you, whether it is
1.d4 Nf6 2.Nf3/g3 or whatever. Still, if White is a weaker player he can exploit the
King's Indian defense to exchange early and try to draw--he has good chances to fulfill this goal as things currently stand.
Queen's gambit declined is also immune to transpositions and has that benefit that its exchange variation is everything but a draw! It leads to some exciting positions and both opponents must know what they are doing. It is highly theoretical but the number of lines is fairly low. They are not razor-sharp but you need to know them or else you will be "smothered to death". Black usually parries White's threats first, and "shoots" later. Most of the time you get the king side attack but it usually draws.
Queen's gambit declined lines without the exchange variation are also theoretical, but the number of lines is relatively small there as well.
If we compare
King's Indian defense and
Queen's gambit declined then KID gives you better winning chances compared to QGD, but you will benefit more as a player from playing QGD since its pawn structure can arise in many other openings ( Nimzo-Indian, Caro-Kann to name just a few ). Also, QGD lines have withstood the test of time while KID is always in some kind of crisis from time to time.
To make the choice between these two, you should see what type of pawn structures you play against
1.e4 and base your choice on that:
If you play Caro-Kann/French defense/Alekhine's defense... then go with the QGD, but if you play the Sicilian/Ruy Lopez/1...e5 in general then go for KID since the pawn structure is similar. That way you will shorten the amount of time you need to learn the opening and its middle-game.
A SMALL NOTE REGARDING YOUR EDIT:
1.d4 d5 2.Nf3 Nc6!? 3.g3!
and "bye bye" Chigorin! You will enter a Catalan if you are lucky or will get smashed by the pressure White's light squared bishop will exert on your center.
If you still need something on Chigorin defense seek games of A.Morozevich. I think he also wrote a book on the Chigorin defense as well-Google and you shall find it!
Still, be prepared to learn another opening as well just in case wHite "kicks you out" of Chigorin with the move order I posted above.
END OF NOTE
This is my Achilles heel as well, and this is what I did to solve it.
If you have questions leave a comment and I will reply.
Hopefully this answer will help you to solve your complex problem.