Recently, the full version of the Eigenstate Thermalization Hypothesis (ETH) has been systematized using Free Probability. In this paper, we present a detailed discussion of the Free Cumulants approach to many-body dynamics within the microcanonical ensemble. Differences between the later and canonical averages are known to manifest in the time-dependent fluctuations of extensive operators. Thus, the microcanonical ensemble is essential to extend the application of Free Probability to the broad class of extensive observables. We numerically demonstrate the validity of our approach in a non-integrable spin chain Hamiltonian for extensive observables at finite energy density. Our results confirm the full ETH properties, specifically the suppression of crossing contributions and the factorization of non-crossing ones, thus demonstrating that the microcanonical free cumulants encode ETH smooth correlations for both local and extensive observables.

14 pages, 10 figures, accompanying paper to Arxiv:2303.00713