Is there an example to demonstrate why 1 ( 1 / 2 ) equals 2?

I know you are looking for something about pizzas but I think fractions in general work better if you work with a definition more akin to the axiomatic definition.
I mean what is ${\displaystyle \frac{1}{\pi }}$ in terms of pizzas?
I would define ${\displaystyle \frac{1}{x}}$ as the number, which when multiplied by x, gives me 1. This works fine for pizzas because, e.g. $2\times \frac{1}{2}=1$ two half slices is the whole.
Now whatever ${\displaystyle \frac{1}{\frac{1}{2}}}$ is, if I multiply it by ${\displaystyle \frac{1}{2}}$ I get one so that
$\frac{1}{\frac{1}{2}}\cdot \frac{1}{2}=\mathrm{1...}$
but this is nothing other than two... (or multiply both sides of the equation on the left by two).

Use ratios.
Pizza will be consumed at a rate of one person per one-half pizza.
$\frac{dP}{d\mathrm{\Pi }}=\frac{1\text{person}}{\frac{1}{2}\text{pizza}}.$
Pizza is delivered at a rate of 1 pizza per day
$\frac{d\mathrm{\Pi }}{dt}=\frac{1\text{pizza}}{1\text{day}}.$
To avoid an excess of pizza, we require at least
$\frac{dP}{d\mathrm{\Pi }}\frac{d\mathrm{\Pi }}{dt}=\frac{1\text{person}}{\frac{1}{2}\text{pizza}}\cdot \frac{1\text{pizza}}{1\text{day}}=2\frac{\text{person}}{\text{day}}.$