There's a Nature article that describes the experiment and the results, http://www.nature.com/nature/journal/v473/n7348/full/nature10104.html, but that's behind a paywall. The experiment is described in some detail in "Prospects for the measurement of the electron electric dipole moment using YbF", http://arxiv.org/abs/1103.1566 (I've only scanned the latter, but it looks to be quite informative).
From the Nature article, "In an atom or molecule with an unpaired valence electron, the interaction of the electron EDM [Electric Dipole Moment] with an applied electric field results in an energy difference between two states that differ only in their spin orientation. This energy difference is proportional to $d_e$ and changes sign when the direction of the field is reversed. A sensitive method of measuring this energy difference is to align the spin perpendicular to the field and measure its precession rate, which is proportional to the energy difference. An alternative description of the method is in terms of an interferometer. There is quantum interference between the two spin states, and the EDM appears as an interferometer phase shift that changes sign when the electric field is reversed."
An electron is not either a sphere or not-a-sphere, but we can introduce more or fewer internal degrees of freedom into the quantum fields that are used to describe experiments results that we attribute to the electron field. Introducing different degrees of freedom has consequences for the geometrical configurations of recorded experimental results. The Nature article is explicit in saying that this is intended to distinguish between different speculative quantum field theories, "many extensions to the standard model naturally predict much larger values of $d_e$ that should be detectable". This is an experimentalists' article, however, so they link to a theory paper on the subject (which I cannot access directly). If these fields give better descriptions than the standard model of particle physics, we expect to see different, less geometrically symmetrical statistics of events.
Many of the problems of reference here can be avoided if we talk about electron fields instead of about electrons. An "electron field" is less likely to be misrepresented as spherical or not spherical, but it can be associated with (representation spaces of) space-time symmetry groups (which describe in a systematic way how something deviates from being symmetrical). Care is needed because a quantum field is a more elaborate mathematical object than a classical field, but we can loosely think of a quantum field as a way to generate probabilities that the configuration of a classical field is one thing or another at any single time, while the details of quantum measurement are such that we can't talk about such probabilities at multiple times.