  We introduce a novel minimally-disturbing method for sub-nK thermometry in a
Bose-Einstein condensate (BEC). Our technique is based on the Bose-polaron
model; namely, an impurity embedded in the BEC acts as the thermometer. We
propose to detect temperature fluctuations from measurements of the position
and momentum of the impurity. Crucially, these cause minimal back-action on the
BEC and hence, realize a non-demolition temperature measurement. Following the
paradigm of the emerging field of \textit{quantum thermometry}, we combine
tools from quantum parameter estimation and the theory of open quantum systems
to solve the problem in full generality. We thus avoid \textit{any}
simplification, such as demanding thermalization of the impurity atoms, or
imposing weak dissipative interactions with the BEC. Our method is illustrated
with realistic experimental parameters common in many labs, thus showing that
it can compete with state-of-the-art \textit{destructive} techniques, even when
the estimates are built from the outcomes of accessible (sub-optimal)
quadrature measurements.
