The burner has a TQFP-80 part which has a very fine pin pitch. The cart has TSOP-32 which is roughly the same pitch. Usually these parts are reflow-soldered in an oven which makes it a trivial process. In my case these are first prototypes and it would be kind of stupid to invest in a stencil which will be obsolete in no time.

So what options are there? I prefer the drag soldering technique. This is done by applying a generous amount of flux to the solder pads, applying a solder bridge to the iron tip followed by dragging the tip along the pins. Surprisingly, it works rather well - I got these parts soldered in no time!

Here's a pretty good video demonstration of the technique done by Sparky's Widgets.