  Motivated by the lack of rotating solutions sourced by matter in General
Relativity as well as in modified gravity theories, we extend a recently
discovered exact rotating solution of the minimal Einstein-scalar theory to its
counterpart in Eddington-inspired Born-Infeld gravity coupled to a Born-Infeld
scalar field. This is accomplished with the implementation of a well-developed
mapping between solutions of Ricci-Based Palatini theories of gravity and
General Relativity. The new solution is parametrized by the scalar charge and
the Born-Infeld coupling constant apart from the mass and spin of the compact
object. Compared to the spacetime prior to the mapping, we find that the
high-energy modifications at the Born-Infeld scale are able to suppress but not
remove the curvature divergence of the original naked null singularity.
Depending on the sign of the Born-Infeld coupling constant, these modifications
may even give rise to an additional timelike singularity exterior to the null
one. In spite of that, both of the naked singularities before and after the
mapping are capable of casting shadows, and as a consequence of the mapping
relation, their shadows turn out to be identical as seen by a distant observer
on the equatorial plane. Even though the scalar field induces a certain
oblateness to the appearance of the shadow with its left and right endpoints
held fixed, the closedness condition for the shadow contour sets a small upper
bound on the absolute value of the scalar charge, which leads to observational
features of the shadow closely resembling those of a Kerr black hole.
