@BrunoStonek if you care about the multiplicative structure beyond E_1 then I don't know of another way. If you only care about the E_1-ring structure, then $BU\times \mathbb{Z}$ is the tensor product of $BU$ and $\mathbb{Z}$ in $\mathsf{Alg}_{\mathbb{E}_1}(\mathsf{Spaces}_{/\mathrm{Pic}})$ so you could just use that the Thom spectrum functor is symmetric monoidal to get the formula $MU \wedge S^0[t]$ as $\mathbb{E}_1$-rings (and hence $MU$-modules)