  For Dirichlet $L$-functions in $\mathbb{F}_q [T]$ we obtain a hybrid
Euler-Hadamard product formula. We make a splitting conjecture, namely that the
$2k$-th moment of the Dirichlet $L$-functions at $\frac{1}{2}$, averaged over
primitive characters of modulus $R$, is asymptotic to (as $\mathrm{deg} R
\longrightarrow \infty$) the $2k$-th moment of the Euler product multiplied by
the $2k$-th moment of the Hadamard product. We explicitly obtain the main term
of the $2k$-th moment of the Euler product, and we conjecture via random matrix
theory the main term of the $2k$-th moment of the Hadamard product. With the
splitting conjecture, this directly leads to a conjecture for the $2k$-th
moment of Dirichlet $L$-functions. Finally, we lend support for the splitting
conjecture by proving the cases $k=1,2$. This work is the function field
analogue of the work of Bui and Keating. A notable difference in the function
field setting is that the Euler-Hadamard product formula is exact, in that
there is no error term.
