If I remember well the Kirszbraun's extension of  a $L$-Lipschitz map $f:U\subset H_1\to H_2$ has the following canonical construction, analogous to the one-dimensional case you mentioned (so in a sense it is explicit). 

Let  $\mathcal{Co} (H_2)$ denote the metric space of all non-empty  bounded closed convex sets of $H_2$ endowed with the Hausdorff distance. Let  $f_*:H_1\to \mathcal{Co}(H_2)$ be defined by $$f_*(x):=\cap_{u\in U}\overline{B}(f(u),L\|x-u\|)$$

(In other words, $f_*$ takes $x\in H_1$ to the set of the admissible values at $x$ for any $L$-Lipschitz extension of $f$ to $U\cup\{x\}$). This map $f_*$ has the same Lipschitz constant of $f$, w.r.to the Hausdorff distance on $\mathcal{Co}(H_2)$.

Any non-empty bounded closed convex $C$ of a Hilbert space $H$ has a well-defined   point $\kappa( C), $ the center of the closed ball of minimum radius containing $C$; this point is unique, and the corresponding map $\kappa: \mathcal{Co}(H)\to H $ is $1$-Lipschitz. *(Warning - here there is an issue; it seems this is not the right selection map; see  Junekey Jeon‘s comment below)*

One can therefore define a canonical $L$-Lipschitz extension of $f$ as $\tilde f:=\kappa \circ f_*$. In case $n=1$, the set $f_*(x)$ is just an interval, its end-points are the inf-convolution you mentioned, and the sup-convolution, and this $\tilde f$ is their arithmetic mean.