1. "Just Look At The Flowers" - The Grove
Being entirely honest, you could very well fill this entire list with moments where characters have had to mercifully put loved ones out of their misery.
The thing is, the vast majority of those moments usually involve killing off someone who has either become reanimated or is on their way to becoming reanimated. In Season 4's The Grove, poor Carol had to make the heartbreaking call to murder a child who had not been bitten, but instead was mentally unstable and had herself committed murder.
That child, of course, was Lizzie, one of two youngsters Carol and Tyreese were looking after.
For Lizzie, this troubled figure had a unique relationship with the undead. Getting upset when walkers were killed, playing tag with these zombies, feeding them, killing her own sister Mika so that she can see her reanimate, and planning to kill baby Judith, it's clear that Lizzie is dangerous and a bit bonkers.
All of this famously led to Carol being left with no choice but to kill Lizzie, informing her to "just look at the flowers" before fatally shooting here in the back of the head.
As shown in the Look at the Flowers episode in 2020, Carol is still hugely traumatised by having executed Lizzie, with her having visions of the decapitated head of Alpha telling her to "look at the flowers".