Shooting with the Veritas LA Jack
Eliot, a little background. There are two parts in your comments, and so I shall reply to both.
I was an early adopter of the Veritas LA Jack (2004). Not long after I restored a Stanley #62, its predecessor, Rob Lee offered me one to compare. That was my introduction to Veritas. Review: http://www.inthewoodshop.com/ToolReviews/VeritasLAJack.html
Along the way I acquired a Stanley #51/52 Chute Board and plane. The plane was fragile and repaired, and when Lie Nielsen proposed making a version of the #51 plane, I told Tom LN I wanted the first one. It duly arrived as a birthday present. Fantastic plane! My review: http://www.inthewoodshop.com/ToolReviews/LN51ShootingPlane.html

I was a very happy camper, and then Veritas began designing their version, and I came to road test it before production, which is how I ended up with one. My review: http://www.inthewoodshop.com/ToolReviews/LVShootingPlane.html

Summary: both shooting planes work on the Stanley #52 chute board, and of course, they would work on any shooting board one makes. I love the traditional looks of the LN, which is an excellent performer, but the performance of the Veritas is just a bit higher. This must not be seen to detract from the LN. I use both planes (and know I am very fortunate to be in this position).
Okay, to the Veritas LA Jack …
There are two methods to use this plane. The first is to build a Hotdog Handle. LN offered one for their LA Jack (#62), but Veritas do not, and so I designed and built one. The plans/method are on my website. Actually, I designed two, one made with a mortice chisel and the other with a tablesaw. Both require a lathe. The tablesaw method is easier: http://www.inthewoodshop.com/ShopMadeTools/Building%20the%20Hotdog%20Mk%20II%20for%20the%20LV%20LA%20Jack%20pics.html

The hotdog places the grip over the blade, and this adds a great deal of control when shooting.
However, if you hold the plane correctly, the hotdog handle is unnecessary. The background to this came when I was invited to a Lie Nielsen Hand Tool Event in Perth in 2011. In this I offered all the opportunity to try out the LN #51 plane (on the Stanley #52 cute board), the LN #9 shooting plane, and a Veritas LAJ. Here is a summary of the second method. Article: http://www.inthewoodshop.com/Furniture/ShootingPlanesCompared.html

The #51/52 combination was in a class of its own. Even a novice could easily produce clean edges without fail. The #51 is captured by the rails of the #52, and this guides the plane so that all the user needs do is push the plane forward.
The advantage of the #9 over the LAJ is twofold: there is better registration for the base of the plane, and this encourages a user to push the plane in the ideal manner. When a novice was given the choice of the #9 and the LAJ, they would inevitably plump for the #9.
However … when the user was taught how to push the LAJ, and/or the LAJ was used with a hotdog handle, very frequently this choice was reversed. The factor here is that the LAJ is potentially more tippy because it has a smaller footprint, it. I recall one person thrusting the LAJ forward .. and bouncing it against the fence at a canted angle.
The correct way (in my opinion) of holding the LAJ (and shooting planes generally), is to exert downforce at a central point while simultaneously exerting low lateral sideforce. One must not attempt to simply push the plane against the sidewall to the shooting board. This will unbalance the plane and cause it to cant over.

Downforce is applied by the thumb directly into the dimpled fingerhole. Sideforce is applied by the four fingertips pushing from under the levercap.
This handhold is quite comfortable and makes the LAJ a practical user without a hotdog. Indeed, there is the potential for someone to grasp a hotdog (whether on the LV or LN LAJ planes) and attempt to push it against the fence from high (and not use their fingers to maintain sideways pressure from low).
Contributing to “tippiness” is the amount of “run up” to the board that is used. Many – both experienced and novices alike – would draw the plane back to the start of the runway, and then push it forward fast in an attempt to create momentum, as if this was necessary to power through the end grain. Shooting in this manner would lead to user losing control of the plane.
What is necessary for control is minimum run up. Place the plane with the blade nearly touching the near edge of the board, and then simply push the plane forward, with even pressure and firmly. Since the shaving removed is very fine, a plane with a sharp blade will cut without much effort. Once this was understood, the extra mass of the LAJ was an advantage over the more stable #9.
Regards from Perth
Derek