Recent literature has explored various ways to improve soft sensors by utilizing learning algorithms with transferability. A performance gain is generally attained when knowledge is transferred among strongly related soft sensor learning tasks. One setting where it is reasonable to expect strongly related tasks, is when learning soft sensors for separate process units that are of the same type. Applying methods that exploit transferability in this setting leads to what we call multi-unit soft sensing. This paper formulates a probabilistic, hierarchical model for multi-unit soft sensing. The model is implemented using a deep neural network. The proposed learning method is studied empirically on a large-scale industrial case by developing virtual flow meters (a type of soft sensor) for 80 petroleum wells. We investigate how the model generalizes with the number of wells/units. We demonstrate that multi-unit models learned from data from many wells permit few-shot learning of virtual flow meters for new wells. Surprisingly, regarding the difficulty of the tasks, few-shot learning on 1-3 data points often leads to high performance on new wells.