Can a magnesium atom combine with just one chlorine atom?
In solid state $\ce{MgCl2}$, can be approximately described assuming fully ionic bond, in which $\ce{Mg}$ atoms lose two electrons to form $\ce{Mg^2+}$, while a $\ce{Cl}$ atoms gain one electron to form $\ce{Cl-}$. To balance the two electrons released by $\ce{Mg}$, you need two chlorine atoms. The electrostatic force of attraction (aka the "ionic bond") binds them, and it has the formula $\ce{MgCl2}$.

Now, you wish to form $\ce{MgCl}$. To maintain charge neutrality, you'll want the $\ce{Mg}$ to lose only one electron. This means the valence shell of $\ce{Mg}$ would still have a single electron left in it. This is a very unstable configuration. 

However, it is incorrect to say that $\ce{MgCl}$ cannot form. A research paper1 has explored the very $\ce{MgCl}$ molecule. Note that the conditions were very drastic, $\pu{500^\circ C}$ for pyrolysis and $\pu{2500^\circ C}$ for vaporization respectively. They ensured "the complete formation of $\ce{MgCl}$ molecular species, since $\ce{Cl}$ was the limiting reactant" as you had correctly identified yourself.

Another paper3 describes the formation of the $\ce{MgCl}$ molecule:


  Excited $\ce{MgCl}$ molecules were produced by two different techniques: (i) at Orsay, they were created by mixing Mg vapor with a flow of $\ce{He/Cl2}$ and excited in a “heated” Schuller-type discharge tube, and (ii) at Waterloo, they were generated by using a copper hollow cathode lamp loaded with $\ce{MgCl2}$ powder.


Note that these techniques are indeed quite "unnatural", in the sense that you might not find $\ce{MgCl}$ naturally, but you can certainly create it through artificial conditions. In fact, $\ce{MgCl}$ is fairly popular otherwise. Several papers2 have already discussed the emission spectrum of the $\ce{MgCl}$ molecule. Some4 also explore its laser cooling. Hence, by no means it is an unknown compound and in fact, can definitely be formed.



References:


Rina Lourena da S. Medeiros, Sidnei de Oliveira Souza, Rennan Geovanny Oliveira Araujo, Djalma Ribeiro da Silva and Tatiane de A. Maranhão, Chlorine determination via MgCl molecule in environmental samples
using high resolution continuum source graphite furnace molecular absorption
spectrometry, Talanta, http://dx.doi.org/10.1016/j.talanta.2017.08.026
Darji, A.B., Shah, N.R., Shah, P.M. et al. Pramana - J Phys (1985) 25: 571. https://doi.org/10.1007/BF02847235
Vervloet, M. Fourier Transform Spectroscopy 2001. https://doi.org/10.1006/jmsp.2002.8514
The Journal of Chemical Physics 143, 024302 (2015); https://doi.org/10.1063/1.4926389