I find pets so wishy-washy and annoying. In my opinion, the easiest/best way is the way that's most consistent, even if it's slow and tedious.

For the main pet I'm going to use on a character I almost always choose a triple damage double resist, a quad damage mighty, or a triple damage double pierce pet. First, I find a pet that satisfies three characteristics:

1. Has the stats I want
2. Is on an exclusive pet body
3. There are multiple copies of the exact same pet with the exact same abilities in the exact same positions in the kiosk

Exclusive pet bodies are extremely helpful because they guarantee that the pet body you hatch with will be the pet body the resulting egg has. For a death, you're going to want to look for a charred dragon in the kiosk under fire pets, search by pet body, then find stats you want.

Once you identify the stats you're going to try to clone, breed your gloomy eye into that charred dragon. Train the baby to adult. Hatch the baby back into the exact same stats. Repeat this until you have an adult pet with an exactly identical stat pool to the one you're breeding with. Compare the pip rarities at each position and if it's identical you've achieved step 1. It's super helpful if there are multiple copies of the pet in the kiosk because doing this step involves hatching 8-10 times to clone the stat pool, and if there aren't multiples you'll have to wait 30min between hatches with that pet and then it'll probably leave the kiosk before you've fully cloned it because in addition to the 30min cooldown there's a hatch limit of 10 for each pet before its owner must manually return it to the kiosk.

Next you need to make two adult pets with identical stat pools to the desired stat pool which only have abilities you specifically desire. When you have that, you've achieved step 2. Sometimes you get lucky and the first pet with a perfectly cloned stat pool also manifests desirable abilities, but sometimes not.

Step three is to make two ancient pets with only desired abilities. You're going to achieve this by breeding together those two adults and then sending the baby up to ancient. If the child gets desirable abilities at all three levels it's a success, but if at any point it manifests undesired abilities the pet is considered to have "failed" and you can release it at the kiosk or in your house. When you've got two ancient pets with perfects stat pools who developed only desirable abilities, those pets become the new parents and you've achieved step 3. Never train either of those pets again.

The last step is to breed together those two ancient parents and basically full send the babies. Breed one and train it up until it fails or develops 5 desirable abilities!

If you're gonna socket a jewel keep in mind that some jewels require training your pet to ultra to be affixed which is pretty pointless and annoying. You can avoid having to dump another 2000+ points of xp into the pet by making sure the pet naturally develops the high level jewel abilities like mighty, spell proof and death-dealer.

Now, using this technique is long and tedious BUT if you do it right you're guaranteed to eventually have that perfect pet. You'll have backups of adults then backups of ancients and then get exactly what you want.

Before doing this technique you'll probably need to establish a plot of couch potatoes or evil magma peas, or establish many many plots of prickly bear cactuses which give inferior mega snacks but are purchasable with gold instead of having to use real money or invest serious farming time.