Driving single 1W LED from 5V
You want a boost converter that can make about 5&frac12; V at 350 mA from 5 V in.  That shouldn't be too hard to find.  At this low voltage and since efficiency matters, get one with synchronous rectification built in.

The more tricky part is arranging the feedback to get the current regulation you want.  At 350 mA, a 1 &Omega; low side current sense resistor would drop 350 mV, which represents about 6% loss.  500 m&Omega; would be less loss, and the resulting 175 mV signal at the current regulation point should be quite good enough.  You can amplify that by a fixed gain to get to whatever regulation feedback voltage the boost chip you pick needs.

Watch out for instability in the boost chip with a opamp in the feedback path.  A boost chip with external compenstation could be useful here.  You could over-compensate the boost chip for more stability.  LEDs don't cause any transients it has to respond to quickly, so no problem there.  Before doing any of that though, try with a ordinary 1 MHz or 10 MHz opamp and see if it works.  Additional compenstation may not be needed.