What is the one-engine-inoperative maximum range flight configuration?
The optimization problem is unchanged from the all engines operative (AEO) cruise optimization. You should still:


Climb as high as practical (colder, thinner air is better for range)
Set the lift coefficient such that drag is minimized for propeller aircraft ($c_{Di} = c_{D0}$). For turbojets, minimize energy per flight time ($c_{Di} = \frac{1}{3}\cdot c_{D0}$). For turbofans, pick a value between those two extremes.


For the computation of the theoretical optimum polar point please see this answer for a complete derivation including wind, or this answer for a shorter explanation.

Your flight altitude will certainly be lower, because you need to generate more thrust on the remaining engine (requiring higher density), which in turn will result in a lower TAS (true air speed). The CAS (calibrated airspeed) should stay about the same, however. The configuration (flaps, gear) should stay unchanged, only rudder deflection will be added for asymmetric thrust compensation. The resulting sideslip will increase the zero-lift drag of the airplane a little, so a slightly lower speed than in the AEO case would result.

In the unlikely case that the required density will result in an altitude below ground level, you will need to slow down and need to set flaps to stay airborne, and now the new optimum with flaps will result in a markedly higher lift coefficient because deploying flaps will increase the zero-lift drag quite a bit. If you have the flaps-down polar, the optimum is easy to find.