Let $C$ be a curve in a closed orientable surface $F$ of genus $g\geq 2$ that separates $F$ into subsurfaces $\widetilde {{F}_{i} } $ of genera ${g}_{i} $, for $i= 1, 2$. We study the set of roots in $\mathrm{Mod} (F)$ of the Dehn twist ${t}_{C} $ about $C$. All roots arise from pairs of ${C}_{{n}_{i} } $-actions on the $\widetilde {{F}_{i} } $, where $n= \mathrm{lcm} ({n}_{1} , {n}_{2} )$ is the degree of the root, that satisfy a certain compatibility condition. The ${C}_{{n}_{i} } $-actions are of a kind that we call nestled actions, and we classify them using tuples that we call data sets. The compatibility condition can be expressed by a simple formula, allowing a classification of all roots of ${t}_{C} $ by compatible pairs of data sets. We use these data set pairs to classify all roots for $g= 2$ and $g= 3$. We show that there is always a root of degree at least $2{g}^{2} + 2g$, while $n\leq 4{g}^{2} + 2g$. We also give some additional applications.