  The very first galaxies that started the cosmic dawn likely resided in
so-called "minihaloes", with masses of $\sim10^5$-$10^8\mathrm{M}_\odot$,
accreting their gas from the intergalactic medium through H$_2$ cooling. Such
molecularly cooled galaxies (MCGs) mostly formed in pristine environments,
hosted massive, metal-free stars, and were eventually sterilized by the
build-up of a disassociating (Lyman-Werner; LW) background. Therefore, their
properties might be very different from the galaxies we see in the later
Universe. Although MCGs are probably too faint to be observed directly, we
could nevertheless infer their properties from the imprint they leave in the
cosmic 21-cm signal. Here we quantify this imprint by extending the public
simulation code 21cmFAST to allow for a distinct population of MCGs. We allow
MCGs to have different properties from other galaxies, including unique scaling
relations for their stellar-to-halo mass ratios, ionizing escape fractions, and
spectral energy distributions. We track inhomogeneous recombinations,
disassociative LW feedback, and photoheating from reionization. After
demonstrating how MCGs can shape the 21-cm signal, we explore to what extent
current observations can already place constraints on their properties. The
cosmic microwave background optical depth from Planck sets an upper limit on
the product of the ionizing escape fraction and the stellar mass in MCGs. When
including also the timing of the putative EDGES absorption signal, we find an
additional strong degeneracy between the stellar mass and the X-ray luminosity
of MCGs. If proven to be of cosmic origin, the timing of the EDGES signal would
have been set by MCGs.
