Congrats! I enjoy my TD-11 very much.. especially with the HD600.. so I hope you'll be quite happy with the pairing, too. Remember to pick up a (male) RCA-to-(female) 1/8" (or 1/4") adapter (i.e. this). This is needed to be able to connect your HD600 to the TD-11's tube output (and the TD-11 doesn't come with such an adapter). The stock tube is very good.. but I recommend reading up about a few of the better 6DJ8/6922/ECC88 tubes and springing for one just for fun. There's quite a few good tubes out there under $50 that punch above their price class. We've discussed a few here and the AudioKarma "Grant Fidelity" subforum has a nice "tube rolling" thread going with reviews & a member maintaining a spreadsheet to keep track of how various tubes (generally) sound in the TD-11 (as commented on by other members). Definitely recommend checking that out.
The TD-11 is my primary desktop DAC and amp that I use with my laptop/desktop (via USB). As I've mentioned before, the only time I use another amp (i.e. the Objective 2 or Leckerton UHA-6S MKII) in conjunction with the TD-11 is if I want to be able to use the TD-11's tube output with highly sensitive phones (i.e. custom IEMs and very efficient headphones which have very low impedance). In this instance, I run a "(male) rca-to-(male) 3.5mm cable" from the TD-11's tube output to the O2 or UHA-6S MKII input, and hook up my efficient headphones or IEMs to it. The O2/UHA acts as a solid state front end to the tube buffer.. and both these portable amps have very low output impedance (<1 ohm). Also, the UHA & O2 are extremely transparent and neutral.. so they won't add any additional coloration to the (already colored) tube buffer.