We revisit the classic broadcast problem, wherein we have $k$ messages, each composed of $O(\log{n})$ bits, distributed arbitrarily across a network. The objective is to broadcast these messages to all nodes in the network. In the distributed CONGEST model, a textbook algorithm solves this problem in $O(D+k)$ rounds, where $D$ is the diameter of the graph. While the $O(D)$ term in the round complexity is unavoidable$\unicode{x2014}$given that $Ω(D)$ rounds are necessary to solve broadcast in any graph$\unicode{x2014}$it remains unclear whether the $O(k)$ term is needed in all graphs. In cases where the minimum cut size is one, simply transmitting messages from one side of the cut to the other would require $Ω(k)$ rounds. However, if the size of the minimum cut is larger, it may be possible to develop faster algorithms. This motivates the exploration of the broadcast problem in networks with high edge connectivity. In this work, we present a simple randomized distributed algorithm for performing $k$-message broadcast in $O(((n+k)/λ)\log n)$ rounds in any $n$-node simple graph with edge connectivity $λ$. When $k = Ω(n)$, our algorithm is universally optimal, up to an $O(\log n)$ factor, as its complexity nearly matches an information-theoretic $Ω(k/λ)$ lower bound that applies to all graphs, even when the network topology is known to the algorithm. The setting $k = Ω(n)$ is particularly interesting because several fundamental problems can be reduced to broadcasting $Ω(n)$ messages. Our broadcast algorithm finds several applications in distributed computing, enabling $O(1)$-approximation for all distances and $(1+ε)$-approximation for all cut sizes in $\tilde{O}(n/λ)$ rounds.