Those controllers (Omega CN7823 and AutomationDirect SL4848VR) are both excellent. They work very well, but are industrial controllers and are not nearly as user-friendly as dedicated kiln controllers when it comes to setting up the programs. They are, as far as I can tell, the same controller. I have used both the AD and Omega versions and buy whichever is cheapest at the time.
They are able to autotune at up to 4 different temperatures and can automatically select the set of PID terms from the autotune temperature closest to the setpoint. This is nice, since it saves having to decide on a single temperature at which to optimize the control.
The "industrial" nature of these controllers means the menu system is not organized to keep the end user away from the settings that can louse things up. Setting the ramp/soak profiles requires you to go past the thermocouple type setting for instance.
I had a maker using an oven I had built inadverdently change the thermocouple type from N to S. The oven only reached tempering temperature but showed Austenitizing temperature as a result. That one was easy to diagnose and fix with a phone call, no harm done, no damage to any workpieces. It could easily have gone the other way though. Since then I've been keeping a book in which all the settings are noted with each oven.
I am not a great fan of type K thermocouples in HT ovens, much preferring type N. Type N was originally developed as an "improved type K" for high temperature applications (above about 1000 degC, 1832 degF) with reduced "drift". Type N tops out at 1300 degC (2372 degF) which is fine for HT ovens, even those intended to handle HSS, though I still use type K for forges, where the higher, 2500 degF, top end is useful for checking forge temperatures.
I have some industrial experience of thermocouples in temperature-controlled combustion plant and type N have proved vastly superior to type K in that application (1000 degC control setpoint, 1832 degF). That said, it's a different application to HT ovens
I tend to use Mineral Insulated type N thermocouple assemblies with either Nicrobell or "Super Omegaclad XL" sheaths and grounded junctions. These give nice fast response and can handle long horizontal insertion depths for accuracy. I usually buy transition joint assemblies with a meter or two of cable and a miniature plug already fitted to eliminate any chance of getting the wiring wrong. I use a miniature panel socket on the control box.
I am currently at the testing stage with a pair of 28" ovens that will bolt together to make one long one. I have stayed with a horizontal arrangement, partly to use a single control box and avoid the problem of vertical temperature gradient.
I'm always interested to know what process other people go through to specify equipment for their needs, so any more information the OP can provide would be appreciated.