Q:

What shape does a "cycloid egg" trace as it rolls on flat surface?

A point on a circle rolling on a flat surface traces a cycloid.

What curve does point on a rolling "cycloid egg" trace? 

I tried to draw what it would trace, but failed.
Also, I think the more interesting question here is what shape with a point on its edge traces a half circle as it roll on a flat surface?

A:

Identify the Euclidean plane $\mathbb{R}^2$ with the complex plane  $\mathbb{C}$.
Given any geometric shape $K$ lying on the upper half plane touching the real axis at origin, nice enough so that following description make sense.
Let $\gamma_K : \mathbb{R} \to \partial K \subset \mathbb{C}$ be a parametrization of $\partial K$, the boundary of $K$, by arc length $s$ subject to the initial condition: $\gamma_K(0) = 0$ and $\gamma_K'(0) = 1$. 
When we roll $K$ along the real axis in the positive $x$-direction, the roulette, the locus of the point
on $\partial K$ originally at origin will be given by the formula
$$z_K(s) = s - \frac{\gamma_K(s)}{\frac{d\gamma_K(s)}{ds}}\tag{*1}$$
As an example, consider the case $K$ is the unit disk $D$. We can arc-length parametrize $\partial D$ as
$$\gamma_D(\theta) = i(1 - e^{i\theta})$$
This corresponding roulette will be a cycloid given by the formula:
$$z_D(\theta) = \theta - \frac{i(1-e^{i\theta})}{e^{i\theta}} = \theta -i( e^{-i\theta} - 1)\tag{*2}$$
When $B$ is a "cycloid egg" $E$ corresponds to a unit circle and the bottom of it is touching the origin, we can use $(*2)$ to work out following parametrization of $\partial E$:
$$\gamma_E(\theta) = 
\begin{cases}
\theta - i(e^{i\theta} - 1), & \theta \in [-\pi,\pi]\\
2\pi - \theta + i(e^{i\theta} + 3),&\theta \in [ \pi, 3\pi]
\end{cases}
$$
For $\theta$ outside $[-\pi,3\pi]$, we can extend this parametrization by periodicity.
Notice
$$\frac{d\gamma_E(\theta)}{d\theta} = 
\begin{cases}
+(1 + e^{i\theta}), & \theta \in [-\pi,\pi]\\
-(1+e^{i\theta}), & \theta \in [ \pi,3\pi]
\end{cases}
= 2\left|\cos\frac{\theta}{2}\right|e^{i\frac{\theta}{2}}
$$
The arc-length $s$ for $\partial E$ is
$$s(\theta) = \int_0^\theta 2\left|\cos\frac{t}{2}\right| dt = 
\begin{cases}
4\sin\frac{\theta}{2}, &\theta \in [-\pi,\pi]\\
8 - 4\sin\frac{\theta}{2},&\theta \in [\pi,3\pi]
\end{cases}
\quad\text{ and }\quad
\frac{d\gamma_E(\theta)}{ds(\theta)} = e^{i\frac{\theta}{2}}
$$
Substitute this back into $(*1)$, we get
$$z_E(\theta) = 
\begin{cases}
2\sin\frac{\theta}{2} - \theta e^{-i\frac{\theta}{2}},&\theta \in [-\pi,\pi]\\
8 - 4\sin\frac{\theta}{2} - (2\pi - \theta + 3i)e^{-i\frac{\theta}{2}} - i e^{i\frac{\theta}{2}},&\theta \in [\pi,3\pi]
\end{cases}
$$
For $\theta \not\in [-\pi,3\pi]$, pick an integer $N$ such that
$\theta_0 = \theta - 4\pi N \in [-\pi,3\pi]$, then periodicity of $\gamma_{E}$ implies
$$z(\theta) = z(\theta_0 + 4\pi N) = z(\theta_0) + 16N$$
In terms of real coordinates, we have
$$(x(\theta),y(\theta)) =
\begin{cases}
\left( 2\sin\frac{\theta}{2} - \theta\cos\frac{\theta}{2}, \theta\sin\frac{\theta}{2}\right), &\theta \in [-\pi,\pi]\\
\left(
8 - 6\sin\frac{\theta}{2} + (\theta-2\pi)\cos\frac{\theta}{2},
-4\cos\frac{\theta}{2} - (\theta-2\pi)\sin\frac{\theta}{2}
\right), & \theta \in [\pi, 3\pi]
\end{cases}
$$
and similarly,
$$x(\theta) = x(\theta_0) + 16N\quad\text{ and }\quad y(\theta) = y(\theta_0)$$
for $\theta = \theta_0 + 4\pi N \not\in [-\pi, 3\pi]$, $\theta_0 \in [-\pi,3\pi]$ and $N \in \mathbb{Z}$.
Following is a picture of the roulette of the "cycloid egg" for $\theta \in [0,4\pi]$.
$\hspace1in$ 
As one can see, qualitatively it is very similar to the ordinary cycloid except
it is flattened at the top. Aside from that, there doesn't seem anything special.

