Clutch

I killed all critters in the area for about 45 minutes before a Clutch finally spawned. Blue, happily. The elite walked over us during the battle but luckily didn't do anything to knock me out of the battle. 

Found a rare quality one within 30 seconds of my arrival. In the pool next to Baron Halis - this big elite that walks in circles

Looks like a build that would be suited towards H. However, the H/S or P/B are likely the best breeds. Making an argument for speed in made by Mummy Wrap, and Tordado Punch with the stun chance.

Found an uncommon H/S the second time I went looking. The first time I netted a rare S/B, but any pet this annoying needs to live longer. The mental anguish this fella causes is profound.

If Clutch is not up, look for a non-combat version of running around (usually spawns / running around in the western corner) and kill it.

Clutch (or another non-combat version) showed up about 2 minutes later.

Found in the House of Constructs, as listed in the other psts.

I found Clutch almost immediately after arriving in the contructs area and captured him. He was the primary so it was easy to see him. Rare quality too, on the first try. He is mine now. Malygos realm 12/8/2020 @10:21pm server time. 

He spawned literally right under the leet. I got him rare (only one there) and then of course after I caught him I got squished by the leet. lol.

Although these might not be guarenteed to spawn as rare, they do seem to have a much higher chance.  The first one I found was rare, so at first I thought I was just lucky. I stayed in the area farming for other pets, and after about 10-15 minutes, I saw another Clutch.  I checked and it happened to be another rare.  I'm not THAT lucky, plus several other people mentioned getting rares on their first try. So if you don't find one the first time, try again before you use a stone.

Found two so far (one spawn at a time tho), one was uncommon H/S, second rare S/B.  Looks like it is, unfortunately, NOT rare only pet. Have plenty of  stones tho so no problem ;). Battlewise very good pet IMO, had few successful battles with him already ;).

As everyone else has said, House of Constucts.  I only saw one of them the whole time I was there and luckily it was rare.  It was right down in the middle with the big roaming boss.

Found as a primary pet near House of Constructs, rare quality