Learn French: Est-ce que vous avez confirmé cette méthode ?

This is the passé composé, the standard past tense in French. It is made of two parts: the helping verb avez (have) and the past participle confirmé (confirmed). Together, they translate to 'did confirm' or 'have confirmed' in English.

Could I ask this question without using est-ce que?

Yes! You can use inversion, which reverses the subject and verb and sounds a bit more formal: Avez-vous confirmé cette méthode ? For everyday spoken French, you can also just use a rising tone at the end of a regular statement: Vous avez confirmé cette méthode ?

Why is it cette méthode and not ce méthode?

The word méthode is a feminine noun. Because of this, you must use the feminine form for 'this', which is cette. If the noun were masculine, you would use ce (or cet if it started with a vowel sound).