The IceCube Neutrino Observatory is a cubic-kilometer Cherenkov detector at the South Pole, designed to study neutrinos of astrophysical origin. We present an analysis of the Medium Energy Starting Events (MESE), which is a veto-based event selection that selects neutrinos from a background of cosmic rays. This is an extension of the High Energy Starting Event (HESE) analysis, which established the existence of high-energy neutrinos of astrophysical origin. The HESE sample is consistent with a single power law spectrum with best-fit index 2.87+0.20-0.19, which is softer than complementary IceCube measurements of the astrophysical neutrino spectrum. While HESE is sensitive to neutrinos above 60 TeV, MESE improves the sensitivity to lower energies, down to 1 TeV. In this analysis we use an improved understanding of atmospheric backgrounds in the astrophysical neutrino sample via more accurate modeling of the detector self-veto. A previous measurement with a 2-year MESE dataset had indicated the presence of a possible 30 TeV excess. With 10 years of data, we have a larger sample size to investigate this excess. We will use this event selection to measure the cosmic neutrino energy spectrum over a wide energy range. The flavor ratio of astrophysical neutrinos will also be discussed.