“Best” is rather subjective (what makes one beneficial gain of function mutation, assuming it is the conjunction of these two attributes you find interesting, better than another?), but there are certainly examples known of mutations that are simultaneously gain of function while also being selectively beneficial. Here’s one:
https://academic.oup.com/mbe/article/37/5/1329/5707441
One mechanism for creating new genes is the gene fusion model that involves reorganization of existing genes, such that fragments of different genes fuse together resulting in novel functionality. Comparative genomics have provided support for the origin of new genes by fusion. Among these, one class includes fusion of entire domains from different proteins that results in the novel protein performing functions of both the domains. Examples of evolution of new genes by this mechanism include the origin of tRNA synthesases ([Berthonneau and Mirande 2000](javascript:;); [Eswarappa et al. 2018](javascript:;)) and fatty acid chain desaturases ([McCarthy and Hardie 1984](javascript:;)). This mechanism for the formation of new genes was also observed during experimental evolution studies in Pseudomonas fluorescens , where fusion of different domains of a desaturase and a di-guanylate cyclase resulted in the generation of an adaptive phenotype ([Farr et al. 2017](javascript:;)). The second class among the gene fusion models includes nonspecific chimeric formation where different fragments of different genes fuse together. Origins of adh-jingwei and adh-twain genes in Drosophila species are examples of this category ([Long and Langley 1993](javascript:;); [Jones et al. 2005](javascript:;)).
A second mechanism of origin of new genes, especially observed in but not limited to bacteria, is from extracellular mobile elements that includes phage DNA and conjugative elements (transposons and plasmids) ([Treangen and Rocha 2011](javascript:;); [Wiedenbeck and Cohan 2011](javascript:;); [Blount et al. 2012](javascript:;); [Jerlstrom Hultqvist et al. 2018](javascript:;)). These mobile elements often result in immediate innovative changes in a one-step genetic event and hence are an important source of generating novelty. Examples of evolution of novel genes by contribution of these mobile elements include the evolution of metabolic pathways ([Pal et al. 2005](javascript:;); [Homma et al. 2007](javascript:;)), diversification of cell-envelope surface structures, synthesis of lipopolysaccharides, and novel regulatory interactions ([Nakamura et al. 2004](javascript:;)).
We describe here an experimental example of an origin of a new gene where both of the above-mentioned mechanisms interplay. Our experiments show how phage DNA when fused with an existing bacterial gene results in novel functionality. More specifically, a chimeric gene is formed by addition of a 169-bp fragment of foreign DNA to a truncated lacI gene. When translated into a protein, due to an internal stop codon, this 169-bp region adds only 23 amino acids to the C-terminal of the truncated LacI protein. When expressed, the chimeric protein can suppress temperature sensitivity in a mutant of Salmonella enterica serovar Typhimurium strain LT2 (designated S . Typhimurium throughout the text) at nonpermissive temperatures. The gene fusion results in relocalization of the chimeric LacI protein to the outer membrane, which results in an increase in membrane vesicle formation and suppression of the temperature-sensitive phenotype. Furthermore, the native repressor functions (i.e., DNA binding and inducer response) of the LacI protein are maintained in the chimeric protein, even though they are not needed for the novel function.
So here’s an example of a mutation, specifically an insertion of a piece of DNA originating from the phage genome, into the reading frame a bacterial protein coding gene, resulting in a novel fusion protein that is 23 amino acids longer, which subsequently is simultaneously capable of maintaining the original function of the pre-mutation bacterial gene, while it gains the biochemical function of being transported to the outer membrane(one biophysical gain of function), results in membrane vesicle formation (another gain of function), which confers increased temperature tolerance (which is a beneficial phenotypic effect).
One insertion mutation to a gene, making it larger, giving it two new biophysical functions, that confers a novel beneficial phenotype.