I will describe the construction of a harmonic measure that reproduces a harmonic function from its Robin boundary data, which is a combination of the value of the function and its normal derivative. I shall discuss the surprising fact that this measure turns out to be (quantitatively) mutually absolutely continuous with respect to surface measure on a wide class of domains that includes the complement of certain fractals. Based on joint work with Guy David, Max Engelstein, Svitlana Mayboroda and Marco Michetti.