So this is not something we would normally say in everyday speech. In fact, I've never heard this before. That said, double-clutching is a maneuver that is sometimes used when driving a manual transmission vehicle. I believe that the character is saying that he has a lump in his throat, or that he is choking up (to put it another way), but because his character is both violent and poetic, the script writer chose to give him a strange way of saying it that makes us think about heavy machinery. I'm not sure exactly why the writer did this, but perhaps he or she wanted to add more of a scary or ominous tone to the words the character speaks, and therefore decided to use terminology from driving that evokes thoughts of gears, heat, oil, and dirt.

"My throat just choked, twice harder than the usual choke, briefly, momentarily."