= Solution
A natural interpretation of the short-wavelength peak is reflected starlight, while the longer-wavelength peak is planetary <thermal radiation>. A reflected spectrum approximately follows the stellar spectrum multiplied by the wavelength-dependent <geometric albedo>; thermal <radiative flux> approximately follows the planet's <Planck function>, modulated by molecular <opacity>. Thus two peaks need not represent two planetary surface <temperatures>.
For a quantitative estimate assume both are broad $F_\lambda$ peaks, reflection has a slowly varying <geometric albedo>, and star and planet have approximately <blackbody> spectral envelopes. <Wien's displacement law> then gives
$$
T_*\simeq\frac{2898\,\mu\mathrm m\,\mathrm K}{0.7\,\mu\mathrm m}
\simeq4140\,\mathrm K,\qquad
T_p\simeq\frac{2898\,\mu\mathrm m\,\mathrm K}{1.5\,\mu\mathrm m}
\simeq1930\,\mathrm K.
$$
The stellar estimate is compatible at order of magnitude with an old, relatively small main-sequence star; equal age does not mean equal <temperature> to the <Sun>. Assume the far-infrared signal is thermal and sufficiently long-wavelength that the <Rayleigh-Jeans law> applies to both bodies. The <planet-star radius estimate in the Rayleigh-Jeans limit> gives
$$
\frac{F_p}{F_*}=\left(\frac{R_p}{R_*}\right)^2\frac{T_p}{T_*},\qquad
R_p=\frac{R_\odot}{2}\sqrt{10^{-3}\frac{4140}{1930}},
$$
and therefore
$$
\boxed{R_p\simeq0.023\,R_\odot\simeq1.6\times10^7\,\mathrm m
\simeq2.5\,R_\oplus.}
$$
This is a small volatile-rich-planet-sized estimate, not a Jupiter-sized one. It is conditional: peaks caused by molecular windows, strongly chromatic reflection, or a spectrum expressed as $F_\nu$ rather than $F_\lambda$ do not support those two Wien estimates. Without $T_p/T_*$, the far-infrared ratio only fixes $R_p^2T_p/T_*$.
For a close-in hot planet around a small star, transit-based atmospheric observations are the natural route if the orbital geometry allows them. <Exoplanet transmission spectra> gain from the small stellar <radius>, with a limb signal scaling as $2R_pH/R_*^2$; they probe composition at the <day-night terminator>. The stated $10^{-3}$ thermal contrast also makes <exoplanet secondary eclipse> measurements a particularly useful route to the dayside <exoplanet emission spectrum>. Resolving such a close-in small planet by <exoplanet direct imaging> is much harder. \b[Transit and secondary-eclipse spectroscopy are favoured for a transiting close-in interpretation; the supplied spectrum alone does not determine the orbital geometry or a unique best technique.]
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