The verb in the given sentence is 'cook'. The word, 'already' indicates that the verb 'cook' has been completed in the recent past. The present perfect tense (has/ have + past participle form of the verb) is used to indicate activities in the immediate past. The past participle form of the verb is 'cooked'. Hence, 'has already cooked' is the correct choice of words. Option B is correct.
Option A: The words, 'is already cooking' is grammatically incorrect. This is because 'is cooking' is in present continuous tense. It cannot be used along with the word, 'already'. Hence, this option is incorrect.
Option C: The verb, 'was cooked' is in past tense and cannot be used along with the word, 'already'. Hence, this option is incorrect.
Option D: The auxiliary, 'did' is used along with the simple present tense form of the verb in order to emphasise a statement. However, it cannot be used along with the verb in past tense (cooked) or with the word, 'recently'. Hence, option D is incorrect.