It is possible to extend the function first to $ Q=[0,T]\times \Omega$ and then to $\mathbb{R}^{N+1}$. The second step is covered by a general statement for anisotropic Besov spaces, theorem 18.5 in [O. V.Besov, V. I.Il'in, S. M.Nikolskii, Integral representations of functions and imbedding theorems].
To extend a function $\varphi\in C^{\frac{k+\alpha}{2};k+\alpha}([0,T]\times\partial\Omega)$ to $C^{\frac{k+\alpha}{2};k+\alpha}(\bar Q)$ one can use a solution of some auxiliary BVP problem for the heat operator $L=\partial_t-\Delta$, since for solutions the required smooothnes is known. First let's extend $\varphi$ to $\tilde\varphi\in C^{\frac{k+\alpha}{2};k+\alpha}([-T,T]\times\partial\Omega)$ for $t<0$ as a linear combination of functions of the form $\varphi(x,-t/i)$, $i=1,\ldots,k+1$. This construction is in effect one-dimensional and is described in Lemma 6.37 in [D. Gilbarg, N. Trudinger. Elliptic Partial Differential Equations of Second Order]. Let $\varkappa\in C^\infty_0(\mathbb R) $ be s.t. $\varkappa(t)=1$ for $t\ge0$ and $\varkappa(t)=0$ for $t\le -T/2$. Put $\hat \varphi=\tilde\varphi\varkappa$. Now the solution of thw BVP
$$\left\{
\begin{eqnarray}
Lu&=&0,\\
u|_{[-T,T]\times\partial\Omega}&=&\hat\varphi,\\
u|_{t=-T}&=&0,
\end{eqnarray}
\right.
$$
belongs to $C^{\frac{k+\alpha}{2};k+\alpha}([-T,T]\times\Omega)$. The compatibility conditions hold because initial and boundary conditions (for $t<-T/2$) are zero.
