we all know and love s-t minimum cut algorithms, but they all cut through the edges in the graph. Are there any variants that cuts through nodes?
See, for example, p. 205 and Theorem 11.7 of Bondy and Murty's Graph Theory with Applications (1976).
"Menger's theorem" is a good keyword for further googling. The usual max-flow min-cut theorem implies the edge-connectivity version of the theorem, but you are interested in the vertex-connectivity version.
(These are existential results, but a typical proof of the vertex-connectivity version reduces the problem to the edge-connectivity version, and then you can apply efficient max-flow/min-cut algorithms.)
It is not very difficult to transform the vertices problem to an equivalent edge version.
Consider a vertex v. Replace v with v1 and v2. Join v1 and v2 with an edge of capacity 1. All the incoming edges into v go into v1 and all the outgoing edges from v, leave v2. ie (u v) -> (u v1) and (v w) -> (v2 w) This is equivalent to edge version of the s-t min cut algorithm.