Statistically, why is the number of mutated genes in eggs treated with chemical mutagenesis one?
The expected number of mutation would not follow normal distribution (ND) as you speculated because a ND would have negative values too as it ranges from $(-\infty,\infty)$, as Remi has also pointed out. In many cases a ND can be approximated because the variance would be so less that you can practically find no negative values. However, it is a wrong assumption to consider all biological statistics as ND by default. Please see this post. Basically, ND approximations can be made on multiple measurements that follow the central limit theorem. Now, in your case of a ND centered at 1, I'll tell you how that is impossible. You can have 2 and more than 2 mutations but you cannot have less than 0 mutations. The distribution is not symmetric and is skewed towards right. You also cannot have fractional mutations and therefore the distribution has to be discrete and not continuous.

Mutagenesis itself can be modelled as a Poisson process. For more information on this please see this post. So the probability of $k$ mutations in a time window, $t$, can be represented by the following equation:

$$P(n=k)=\frac{(\lambda t)^ke^{-\lambda t}}{k!}$$

Where $\lambda$ is the rate of mutagenesis.

The mean number of mutations (and also the variance for a Poisson distribution) would be $\lambda t$.

Now, the rate of mutagenesis would be proportional to the concentration of the mutagen. I speculate that the relationship between them would follow a saturation kinetics (like Michaelis-Menten) but at lower concentrations it would be essentially linear.

Moreover, the rate of mutagenesis would also depend on the cell type.

Therefore, standardization is required and labs working in this area for a long time develop standardized protocols (with specific mutagen concentrations and treatment times) which would let them avoid multiple mutations. Even then, they have to screen a lot of individuals to filter out the non-mutants. 

Basically, there is no magic number.