If you are assuming the use of the bi-invariant metric, then the geodesics are right/left translations of the one parameter sub-groups $O(t)=O(0) \exp(tG)$ where $G \in \mathfrak{so}(3)$ (i.e. it is anti-symmetric and traceless) and $O(0)\in SO(3)$.
The curve you want will have $O(0)=M_1$ and thus be of the form $O(t) = M_1 \exp(tG)$ with $G=\log(M_1^{-1}M_2)$ (the matrix log). You need to choose a specific matrix log as generically there will be many. The curve is parameterised by $t \in [0,1]$, it is smooth, is a geodesic of the aforementioned metric and has the end points you hoped for.