DisclaimerIn the process of typing up this question, I determine its solution.  Since I went through the trouble of typing up the question in its entirety, I will post its answer as well.  It may help out others who find themselves in the same predicament.  Think of this as a sort of blog-post, if you will.The GoalConsider the mixed boundary value problem\begin{align}\frac{d}{dx}\left(k(x)\frac{du}{dx}\right)=f \text{ in }\Omega\\u=P \text{ at } x=0\\\frac{du}{dx}=T \text{ at } x=1\end{align}where $P$ and $T$ are constants and $f$ is a source term.I'm using finite differences and my goal is to impose the boundary condition at $x=1$ in such a way as to achieve second order accuracy.  Assume that the grid has $N+1$ equispaced points (including the boundary points) given as $x_0,x_1,...,x_N$My ApproachAt the right boundary, use a 2nd order centered difference for the boundary condition\begin{equation}\frac{u_{N+1}-u_{N-1}}{\Delta x}=T\end{equation}and the 2nd derivative operator as:\begin{align}\frac{k_{N+\frac{1}{2}}\frac{u_{N+1}-u_{N}}{\Delta x} -k_{N-\frac{1}{2}}\frac{u_{N}-u_{N-1}}{\Delta x}}{\Delta x}=f_i\end{align}We can solve for the ghost point $U_{N+1}$ in the 1st equation, substitute it into the 2nd equation and simplify.The ProblemUsing this discretization requires the evaluation of $k_{N+\frac{1}{2}}=k(x_N+\frac{\Delta x}{2})$, which is outside of the domain!  In general, $k(x)$ is only defined within the domain and I can't/shouldn't use values outside of it.  Thus, I don't think this is the right approach to achieve 2nd order accuracy.  What else can I do in this case?
Finite Difference Method Neumann Boundary Condition with Variable Coefficients
finite difference;boundary conditions