生气 -> 生我的气 . 的 in a certain grammatical structure
生气 is a verb for "get angry; take offence"

'的' in  生我的气 is an 'adjectival particle' that turns the noun '我'(I) into  '对我的'(toward me) 

'生气' (get angry) --> '生(对)我的(怒)气' = ( 生气 *at me**) = (get angry at me) 

'生气' is a two characters verb, but '气' in '生我的气', is treated as a noun '怒气'

more example:

'发火' (get mad) -->'发不必要的火' = (发 unnecessary 火) = (get mad unnecessarily ) 

'发火' is a two characters verb, but '火' in '发不必要的火' is treated as a noun '怒火'

A more telling example:


  问话 [ask word] = questioning
  
  问你的话 [ask your word] = questioning you
  
  '话' here is treated as a noun 'words' instead of part of the two characters word 问话 (questioning)