Building on research on teacher noticing, the goal of this study was to understand what and how mathematics teacher educators notice critical events and how they make connections to consider the characteristics of distinguished coach noticing, meaning the noticing we would hope those coaching would attain to support teachers. We interviewed 29 mathematics teacher educators in two different experience groups and asked them to respond to vignettes of coach--teacher interactions. We modified an existing analytic framework from teacher noticing research for the coaching context, deeming it the Mathematics Teacher Educator Noticing Framework. Results (a) provide characteristics of distinguished noticing for WHAT mathematics teacher educators notice, with a focus on noticing as related to critical events and (b) describe characteristics of distinguished noticing for HOW mathematics educators notice, with a focus on connecting critical events and principles of learning and/or coaching along with principles in distinguished noticing involving coaching discourse. We describe the differences between two different coach populations--mentor coaches and participant coaches. The findings describe what constitutes critical events for mathematics teacher educator noticing and describe distinguished noticing by illuminating the connections made between critical events (what is noticed) and connections between critical events and principles of teaching and learning (how noticing occurs). Further, findings revealed differing quantities of relationships between teacher thinking and/or coaching actions, suggesting variability in how distinguished mathematics teacher educator noticing occurs.