In this talk, we will explore a Brownian perturbation of the supercooled Stefan problem which arises naturally as a model of contagion in financial markets, conditional on the market risk that remains after full diversification. In line with the work of Delarue, Nadtochiy, and Skholnikov, we first present a probabilistic formulation of the problem. Starting from there, we then give a simple outline of a succinct argument for whether or not blow-ups will occur with probability one, depending on the initial specification. Finally, we turn to the financial application, looking at how the problem may serve as the basis for a forward-looking systemic risk indicator, aiming to capture the current market sentiment, and briefly discussing ongoing empirical work in this direction.