Preposition followed by participial phrase

Solution 1:

According to CGEL (p. 641), after brushing my teeth is a preposition phrase. Its head is the preposition after, while the gerund-participial clause brushing my teeth is its complement.

Other types of clauses can also be complements of prepositions. Here is the relevant section of CGEL; the relevant example is [19vii]:

5 The structure and functions of PPs

5.1 Complementation

(e) Complements with the form of clauses

[19]   i  This happened [after Stacy left].                                       [non-expandable declarative]
         ii  I'll do it [provided that you help me].                                     [expandable declarative]
        iii  [Although (we were) nearly exhausted,] we pressed on.             [reducible declarative]
        iv  Let me repeat [lest there be any doubt about the terms].                     [subjunctive]
         v  They ignored the question [of whether it was ethical].                           [interrogative]
        vi  We can't agree [ on how much to charge].                             [infinitival interrogative]
       vii  They're talking [about moving to New York].                                [gerund-participial]
      viii  He's not as enthusiastic [as he used to be].                                                 [comparative]

Non-interrogative infinitivals are found with the expressions in order and so as: He only mentioned it [in order to embarrass his wife] or We left at dawn [so as to miss the rush-hour traffic]. Constructions with clausal complements are dealt with in the context of our discussion of subordinate clauses in Chs. 11,13, and 14. Note that in They kept blaming him [for what he had done] the underlined constituent is not a clause but an NP (a fused relative), and hence does not belong under the present heading.