"que mamie avait faite exprès pour moi" - where does the faite come from? Why is there an e?

‘Que’ is the direct object pronoun preceding the verb - replacing the noun la jupe. 

A case of the past participle agreeing with the direct object pronoun preceding the verb with auxiliary avoir in compound tenses (noting it does not only apply to passé composé but all compound tenses with avoir auxiliary.  Aurélie - perhaps a note on this could be added to the lesson linked below that uses passé composé to demonstrate the principle ?). 

Special cases when the past participle agrees (in number & gender) when used with 'avoir' in the compound past in French (Le Passé Composé) 

Que = Whom/which/that (French Relative Pronouns)

The explanation of whether the participle agrees in number and gender is usually encapsulated in several rules with accompanying exceptions. For example, it agrees with the auxiliary verb être and it doesn‘t agree with the auxiliary verb avoir. Except in some cases, where it does‘t agree with être and it does agree with avoir.  Actually, there‘s only ONE SINGLE rule, I believe: it always agrees when the COD comes before the participle, regardless of whether the auxiliary verb is avoir or être. Without exception…..that I can think of. 

Chris, this ‘rule’ is not comprehensive, and incorrectly suggests that être verbs also take agreement from the direct object, rather than from the subject of the verb. 

Être verb conjugations only ever ‘agree’ with the subject, never specifically with the object.  However, it is true that agreement with the subject with pronominal verbs only occurs when the reflexive pronoun is a direct object, and not when it is an indirect object.

Avoir verbs never ‘agree’ with subject, and  only ever have agreement with a direct object that precedes the verb.  ( And then there is a special ‘exception’ with verbs of perception that require the subject to precede the infinitive verb as well ). 

In compound tense conjugations :

1. être only agrees with the subject, but doesn’t always agree

2. Avoir only ever agrees in compound tenses with a direct object preceding the verb ( and with verbs of perception only if the subject precedes the infinitive verb ). 

I think if all the nuanced and complex rules of French grammar had simpler explanations, the bilingual native speakers would surely be teaching them. 

https://www.lawlessfrench.com/grammar/verb-agreement/ 

https://www.frenchtoday.com/blog/french-verb-conjugation/passe-compose/ 

https://www.lawlessfrench.com/grammar/agreement-with-pronominal-verbs/

Thanks for the detailed reply, Maarten. I may be daft, I realize, but I'm still wondering how you distinguish that the être verbs agree with the subject and not the reflexive pronoun. For example:

Nous nous sommes rencontrés. -- We ran into each other.

First nous is the subject, second nous is the COD, right?

Nous nous sommes brossé les dents. -- We brushed our teeth.

First nous is the subject, second nous is the COI (indirect object), the COD is les dents.

J'ai les livres, que tu m'as prêtés. -- I have the books that you lent me.

Que (=les livres) is the COD.

On further reading, Chris, I accept it is even more complex than I deduced from the sites I initially linked. However, all sites I have come across still start from the basis that with être, agreement is with the subject, but there are exceptions. They deal with those exceptions differently.

The longer, more complex explanations are consistent with there not being one simple rule.  It only applies to être in some cases, not in all situations. It covers avoir in virtually all, except perhaps the special case Laura Lawless notes with verbs of perception having a slightly different criterion.

Agreeing past participle with subject's gender and number with (+ être) verbs in the compound past in French (Le Passé Composé)

 http://www.french-linguistics.co.uk/grammar/past_participle_agreements.shtml

Maarten, can you give a few examples, please? I looked at the links you provided and couldn't really find one that specifically contradicts what I wrote. I only found corollaries of the examples I quoted in my post above.

You say (and so do the links) that with être-verbs the participle agrees with the subject. Sure, there is no COD in these cases, e.g., je suis parti(e). That's not the class of participle agreement I'm talking about.

With reflexive verbs, the reflexive pronoun can be viewed as a COD. Since it precedes the participle, the participle will have to adjust accordingly:

Je me suis habillé(e). -- (lit.) I am dressing myself. The "myself" can be viewed as a formal COD.
Je me suis brossé les dents. -- Here the COD is les dents and the reflexive pronoun is a COI.

Again, I'd love to get to the bottom of this. At the moment, I don't see how the links you post contradict what I wrote.