What is the smallest uniquely hamiltonian graph with minimum degree at least 3?

I am not sure what the smallest such graph is, but since you also asked for more information on uniquely hamiltonian graphs with minimum degree $3$, Entringer and Swart proved the following nice theorem.

For each $n= 2k, k \geqslant 11$, there exists a uniquely hamiltonian graph on $n$ vertices having two vertices of degree $4$ and all others of degree $3$.

The system encouraged me to answer my own question, although it feels a bit strange to do so.

Anyway, after a bit of thinking and a (more substantial) bit of computing, I can now safely conclude that this 18-vertex 28-edge graph is the smallest uniquely-hamiltonian graph with minimum degree 3, and there are no others of this order (number of vertices) and size (number of edges).