In this paper, we bring the main tools of the Laplacian paradigm to the Broadcast Congested Clique. We introduce an algorithm to compute spectral sparsifiers in a polylogarithmic number of rounds, which directly leads to an efficient Laplacian solver. Based on this primitive, we consider the linear program solver of Lee and Sidford (FOCS 2014). We show how to solve certain linear programs up to additive error $ε$ with $n$ constraints on an $n$-vertex Broadcast Congested Clique network in $\tilde O(\sqrt{n}\log(1/ε))$ rounds. Using this, we show how to find an exact solution to the minimum cost flow problem in $\tilde O(\sqrt{n})$ rounds.