Understanding the generality of the NER problem
The problem described is not a more general version of Named Entity Recognition, it is a different problem called parsing. Parsing consists in extracting the syntactic structure of a text, usually in the goal to better capture its semantics. There are various approaches:

Shallow parsing only identifies the constituents of the sentences (based on your example this could be sufficient in your case)
Statistical parsing and in particular Dependency parsing represent the full structure of the sentence, including the links between its constituents.

There are various libraries and datasets for parsing: one of the most famous is probably the Stanford parser, but there are many others often included in NLP toolkits such as OpenNLP. The Universal Dependencies project is a vast multilingual collection of annotated text which can be used to train parsers.
Semantic Role Labeling (SRL) is a closely related task which consists in identifying the semantic relations between a predicate (verb) and its related constituents (e.g. subject, object).