  The Parker Solar Probe (PSP) and Solar Orbiter missions are designed to make
groundbreaking observations of the Sun and interplanetary space within this
decade. We show that a particularly interesting in situ observation of an
interplanetary coronal mass ejection (ICME) by PSP may arise during close solar
flybys ($< 0.1$~AU). During these times, the same magnetic flux rope inside an
ICME could be observed in situ by PSP twice, by impacting its frontal part as
well as its leg. Investigating the odds of this situation, we forecast the ICME
rate in solar cycle 25 based on 2 models for the sunspot number (SSN): (1) the
forecast of an expert panel in 2019 (maximum SSN = 115), and (2) a prediction
by McIntosh et al. (2020, maximum SSN = 232). We link the SSN to the observed
ICME rates in solar cycles 23 and 24 with the Richardson and Cane list and our
own ICME catalog, and calculate that between 1 and 7 ICMEs will be observed by
PSP at heliocentric distances $< 0.1$ AU until 2025, including 1$\sigma$
uncertainties. We then model the potential flux rope signatures of such a
double-crossing event with the semi-empirical 3DCORE flux rope model, showing a
telltale elevation of the radial magnetic field component $B_R$, and a sign
reversal in the component $B_N$ normal to the solar equator compared to field
rotation in the first encounter. This holds considerable promise to determine
the structure of CMEs close to their origin in the solar corona.
