When I see it correctly in the top figure, there is some " bunching" in your data, meaning that there are a number of companies with the same (or very similar) number of employees. Since you appear to run a regression with only one independent variable (right hand side), this bunching will be visible in the residual.
Your model is:
$$ y_i = \beta_0 + \beta_1 x_i + u_i$$
Now say $\beta_0 = 1$ and $\beta_1 = 0.1$, and with "bunching" in $y$ you will get something like:
y x y_hat u_hat
5 10 2 3
5 20 3 2
5 30 4 1
5 40 5 0
So the linear nature of the model will be reflected in the residual given that there is "bunching" in the data.
Note that using a log-log approach (which is perfectly okay) will change the natural interpretation of the estimated coefficients. In a log-log case, you will interprete the results as "a one percent change in $x$ will be associated with a $\beta_1$ percent change in $y$" (all other things equal).
Since the number of emplyees is bounded (no negative employees -> "count data"), things like Poisson regression could be worth a try.
Overall your model may be underspecified. If you can, include additional $x$-variables in your model, so to better reflect the data generating process. See page 114 in this book.