The balanced connected $k$-partition problem (\textsc{bcp}) is a classic problem, which consists in partitioning the set of vertices of a vertex-weighted connected graph into a collection of~$k$ classes such that each class induces a connected subgraph of \emph{roughly} the same weight. In this study, we investigate edge-weighted variants of $\textsc{bcp}$, where we are given a connected graph $G$, $k \in \Z_\ge$, and an edge-weight function $w \colon E(G)\to\Q_\ge$, and the goal is to compute a spanning $k$-forest~$\mathcal{T}$ of $G$ (i.e., a forest with exactly $k$ trees) that minimizes the weight of the heaviest tree in~$\mathcal{T}$ in the min-max version, or maximizes the weight of the lightest tree in~$\mathcal{T}$ in the max-min version. We show that both versions of this problem are $\NP$-hard on complete graphs with $k=2$, unweighted split graphs, and unweighted bipartite graphs with $k\geq 2$ fixed. Moreover, we prove that these problems do not admit subexponential-time algorithms, unless the Exponential-Time Hypothesis fails. We focus on the min-max version and devise a tight $k$-approximation algorithm, compact and non-compact integer linear programming formulations, branch and cut, and branch and price algorithms. Finally, we present the outcomes of an experimental study on the performances of different solution methods. The source code of the complete implementation of the proposed algorithms is also available.