When you say "burning to QuickTime", I'm going to assume you want to know the best possible way to export out of FCE to create a self-contained QuickTime file for playback on a Mac/PC hooked up to these monitors? What is your source media? Standard Definition DV? HDV? AVCHD? etc...
I haven't used FCE(I'm a FCP user), but I can't imagine the exporting process being any different. If you want the highest resolution possible(providing this is your on-line edit and it's not Uncompressed) for playback on a computer with a standard hard drive, simply File->Export->QuickTime Movie; This'll export your sequence in its native form.
If you need a smaller file size, yet want to retain a high quality, then your option would be to use: File->Export->Using QuickTime Conversion.
In the pop-up window, select the Options button, which will bring up a Movie Settings window. Select the Settings tab and change the Compression type to H.264, keyframe at whatever your frame rate is, and increase the Quality to Best; You can also restrict the data rate if this is destined for web.
Select the Size button and alter the dimensions for your intended output. You can compress the audio instead of leaving it uncompressed as Linear PCM if you want(AAC should be fine).