  We report on our study of high-energy properties of two peculiar TeV
emitters: the "extreme blazar" 1ES 0347-121 and the "extreme blazar candidate"
HESS J1943+213 located near the Galactic Plane. Both objects are characterized
by quiescent synchrotron emission with flat spectra extending up to the hard
X-ray range, and both were reported to be missing GeV counterparts in the
Fermi-LAT 2-year Source Catalog. We analyze a 4.5 year accumulation of the
Fermi-LAT data, resulting in the detection of 1ES 0347-121 in the GeV band, as
well as in improved upper limits for HESS J1943+213. We also present the
analysis results of newly acquired Suzaku data for HESS J1943+213. The X-ray
spectrum is well represented by a single power law extending up to 25 keV with
photon index 2.00+/-0.02 and a moderate absorption in excess of the Galactic
value, in agreement with previous X-ray observations. No short-term X-ray
variability was found over the 80 ks duration of the Suzaku exposure. Under the
blazar hypothesis, we modeled the spectral energy distributions of 1ES 0347-121
and HESS J1943+213, and derived constraints on the intergalactic magnetic field
strength and source energetics. We conclude that although the classification of
HESS J1943+213 has not yet been determined, the blazar hypothesis remains the
most plausible option, since in particular the broad-band spectra of the two
analyzed sources along with the source model parameters closely resemble each
other, and the newly available WISE and UKIDSS data for HESS J1943+213 are
consistent with the presence of an elliptical host at the distance of
approximately ~600 Mpc.
