How do genes determine facial features?
First, it's been estimated that the narrow-sense heritability of facial morphology is about ~60-90% (Martinez-Abadias et al 2009, Weinberg et al 2013, Liu et al 2012), so genetics is likely to be a greater influence than environment. But actually, the molecular genetic understanding of cranio-facial morphology is quite poor relative to other quantitative traits.
There's a list of genes from Wu et al (2019) which have been implicated and replicated across studies, but it's likely that there is a substantial amount of population specific differences in the effect size of different genes. The most replicable gene is PAX3 (which encodes a transcription factor).
As for the actual molecular genetics, it's complicated. Here are some quotes from a review (Weinberg et al 2018):

As with most common traits [61], the implicated variants from the
above facial GWAS tend to reside in noncoding (regulatory) regions of
the genome. Many variants at the 15 aforementioned facial loci were in or near
enhancers known to play a role in human craniofacial development.


An important finding from this study was that the effects of genetic
variants on facial morphology tend to be highly specific, even within
a single structural component like the nose. Another important finding
was that variants at these 15 loci tended to show preferential
activity in human cranial neural crest cells—a critical cell
population for building the face.

As for a more specific example, Adhikari et al (2016) peformed a GWAS on admixed Latin Americans and identified a significant association in the EDAR gene (well studied and known for being at high frequency in East Asian populations and influencing ear wax type, follicle/gland type and teeth formation). EDAR has a general role in the development of ectodermal tissues (such as the skin).
They found that the variant in EDAR was associated with reduced chin protrusion. Other studies have also shown that EDAR variants influence the cranial morphology in mice (using knockout studies). Functionally, they propose that:

The impact of the Eda pathway on mandibular morphology has been
interpreted as resulting from epithelial–mesenchymal interactions
during mouse craniofacial developme


References

Martinez-Abadias, N. et al. Heritability of human cranial dimensions: comparing the evolvability of different cranial regions. J. Anat. 214, 19–35 (2009).
Weinberg, S. M., Parsons, T. E., Marazita, M. L. &amp; Maher, B. S. Heritability of face shape in twins: a preliminary study using 3D stereophotogrammetry and geometric morphometrics. Dent. 3000, 1 (2013).
Liu, F. et al. A genome-wide association study identifies five loci influencing facial morphology in Europeans. PLoS Genet. 8, e1002932 (2012).
Wu, W., Zhai, G., Xu, Z. et al. Whole-exome sequencing identified four loci influencing craniofacial morphology in northern Han Chinese. Hum Genet 138, 601–611 (2019). https://doi.org/10.1007/s00439-019-02008-6
Weinberg, Seth M., Robert Cornell, and Elizabeth J. Leslie. &quot;Craniofacial genetics: Where have we been and where are we going?.&quot; (2018): e1007438.
Adhikari, Kaustubh, et al. &quot;A genome-wide association scan implicates DCHS2, RUNX2, GLI3, PAX1 and EDAR in human facial variation.&quot; Nature communications 7.1 (2016): 1-11.