I am having trouble proving the following statement:
For any smooth function $h\in \mathrm{C}^\infty(\mathbb{T}^n)$ and a function $\psi\in H^{1/2}(\mathbb{T}^n)$, then, there is a unique variational solution $\varphi$ to the problem $$ \Delta_{x,y} \varphi = 0 \quad \text{in}\;\Omega$$ where $\Omega = \{(x,y)\in \mathbb{T}^n\times \mathbb{R}: y < h(x)\}$ with the boundary condition $$ \varphi(x,h) = \psi(x) $$ I cannot picture how this fit into the classical normal Laplace equation framework with this domain, also the classical existence given $H^{1/2}$ data. I would be thankful if you can point me to the source to learn about it.
Given that $\Omega$ is a smooth Riemannian manifold such that $\bar \Omega$ is compact and the boundary $\partial \Omega$ is smooth, there exists a unique weak solution $\varphi \in H^1(\Omega)$ to the boundary value problem $$\left \{\begin{array}{r@{ \ }l @{\quad \;} l} \Delta \varphi &=0 &\text{in } \Omega \\ \varphi &=g &\text{on } \partial \Omega \end{array} \right . \tag{$\ast$} $$ where $g\in H^{1/2}(\partial \Omega)$ and $\Delta$ is the Laplace-Beltrami operator on $\Omega$. A great book on this is Partial Differential Equations I: Basic Theory by M. Taylor, in particular, see Chapter 5. When $\partial \Omega$ is a Lipschitz manifold (which it is here), a definition of $H^{1/2}(\partial \Omega)$ is that it is just the image of the trace operator $T: H^1(\Omega) \to L^2(\partial \Omega)$, (see wikipedia particularly, the section 'Image of the trace operator') so from this viewpoint, $H^{1/2}(\partial \Omega)$ is pretty much defined to be the space for which ($\ast$) is uniquely solvable.
In your case, $\partial \Omega = \{ (x,y) \in \mathbb T^n \times \mathbb R \, \vert \, y=h(x) \}$. Then each $g: \partial \Omega \to \mathbb R$ can be identified with a function $\tilde g : \mathbb T^n \to \mathbb R$ via $\tilde g(x) = g(x,h(x))$. In this way, I think you should be able to show that $H^{1/2}(\partial \Omega) = H^{1/2} (\mathbb T^n)$, but I didn't check this explicitly myself. Thus, the statement you quote is just a (slight) reformulation of existence of weakly harmonic functions with Dirichlet boundary conditions on a particular manifold.