The Nanny/Heartwarming

This show has many, but one of the most notable occurs in "I Don't Remember Mama", when the Sheffields and Fran sit down to watch old home movies with their deceased mother Sara. Earlier in the episode, Gracie had complained that she couldn't remember her mother, so while watching, she sees her infant self playing with her mother, and Gracie exclaims "I remember that!".
Early in season one, C.C. told Gracie that Fran only pretended to love her because of her job. Fran straightened her right out with one of the best speeches ever, all with a charming smile on her face:

Fran: You're a lovely woman and I wish you well...but if you ever hurt one of my kids again, they'll be wiping your blue blood off the walls. And I mean that in the nicest possible way.

- Don't forget Fran's conversation, reassuring Gracie that she really does love her:

Fran: Now, it is true your father pays me to take care of you. But he doesn't pay me for loving you. And I do.

Maxwell's proposal to Fran.
Grace's poem to Fran when she thought she was leaving to marry her ex-boyfriend Danny.

Grace: F is for the fun we have together. R is for the rummy that we play. A is for the answers to my questions. N is for the nasal things you say.

I am surprised this hasn't been mentioned yet. When Maxwell decides to have Fran sign the adoption papers, instead of a pre-nup.
In "Not Without My Nanny", Maxwell travels halfway around the world to tell Fran he loves her.
In Kindervelt Days, Fran is depressed at the prospect of attending a summer camp reunion, so the Sheffield kids sends in her old "Win a date with Eric Estrada" sweepstakes form to cheer her up. Eric actually responds to the form, and offers to be her date to the reunion. However, he has to leave early, so Maxwell steps in as Fran's date.
In "The Butler, the Husband, the Wife and Her Mother", Maxwell and Niles swap roles due to a lie of Fran's. Halfway through the charade, the Butlers' Association drops by to make a surprise appraisal of how Niles is doing as a butler. Maxwell and the rest of the household keep up the charade, but of course things eventually fall apart. The Association representatives are getting ready to give Niles a horrible appraisal, when Maxwell points out that if a family is willing put on that kind of performance for their butler, he must be a truly spectacular butler indeed and that Niles is, basically, family.
In the finale, when Fran's father sings "Time After Time" to Fran and Maxwell at their anniversary.
In the episode where Maxwell's brother, Nigel, proposed to Fran and she was leaving the house for what she thought was the last time. She finds Niles sleeping on the couch and she gives him a kiss on the cheek. Considering that he is one of her best friends, I could only imagine sad them saying goodbye would've been had he been awake.
- Luckily, she comes back

Maxwell: Don't you worry, Miss Fine, I've got you.
Fran: Boy, do you ever

C.C. promises to pay Niles if he escorts her to an event her ex-boyfriend was going to take her to. After their not-date, they dance together, laughing and, for once, legitimately, though still gloriously snarkily, enjoying one another's company. She gives him the check, and he casually rips it up, declaring, "This one's on me," before continuing to dance with her. I think even non Niles/C.C. shippers would agree that was a genuinely sweet scene between the two.

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