I spend the next couple of days shadowing the dialogue over and over and drilling those flashcards.
This definition of drill can be found in the Oxford Dictionary under drill 1, noun:
2.1 Intensive instruction or training in something, typically by means of repeated exercises.
‘tables can be mastered by drill and practice’
count noun ‘language-learning drills’
drill - Oxford Dictionaries
Although a noun, I believe this is an example of a noun spontaneously being used as a verb. In the context of self teaching or self study, it means to perform a drill (a repeated exercise) on or about something:
Sally drilled the multiplication tables.
= Sally did drills on the multiplication tables.
= Sally did multiplication table drills.
Drill the calculus problems so that you will do well on the exam.
= Do drills on these calculus problems ...
I drilled the flashcards.
= I did drills on the flashcards.
Drill as a verb also appears in dictionaries, but with a different usage. Here, it used to mean that someone instructs someone else by means of a drill (a repeated exercise):
2.2 Instruct (someone) in something by the means of repeated exercises or practice.
‘I reacted instinctively because I had been drilled to do just that’
It is possible that the self-study sense of the verb drill (drilling multiplication tables, flashcard, and so on) was not common enough to warrant a separate dictionary entry.