There's still ambiguity as to whether it's:

   (pan-roasted oysters) with (a tomato sauce over rice)

or:

   (pan-roasted oysters with a tomato sauce) over rice

I don't think there is. It is the second. The clear convention is that each meal item is a list element.

It's not clear to me.

> (pan-roasted oysters with a tomato sauce) over rice

I picture a bowl of rice, then oysters on top of the rice, and then tomato sauce poured on top of the oysters.

> (pan-roasted oysters) with (a tomato sauce over rice)

I picture a plate. On one half is oysters, the other half is rice with tomato sauce poured on it.

As someone who would never put tomato sauce on either oysters or rice, either are equally likely.

No, it's a list of two subsets contained in the set "meal."

     - main-dishes (x1)

     - side-dishes (x2)

The entree set has 1 item in it, the "pan-roasted oysters with a tomato sauce over rice.The ingredients of "entree" follow the same hierarchical structure as the ingredients of "meal," so:

     -main-ingredients (x1)

     -side-ingredients (x2)
 

Since we're only talking about two sets here, a comma separating them is unnecessary. You could use one if you wanted, ("pan-roasted oysters, with a tomato sauce over rice") but then the syntax of the sentence would be off. You'd have a list of three things:

     -pan roasted oysters

     -a tomato sauce over rice

     -broccoli salad and bread pudding with chocolate sauce

This is logically inconsistent because the oysters, tomatoes, and rice comprise a single dish. The tomatoes and rice don't stand on their own but describe how the oysters are prepared and served, just as the non-entrees describe how the meal is served.

Lets compare:

ORIGINAL: >“The next day, I enjoyed pan-roasted oysters with a tomato sauce over rice, broccoli salad and bread pudding with chocolate sauce."

OXFORD COMMA: > “The next day, I enjoyed pan-roasted oysters, with a tomato sauce over rice, broccoli salad, and bread pudding with chocolate sauce."

By structuring the sentence in that way, you erase any information about the value of the meal, and the relationships between its components. In return you ensure that someone won't think there was chocolate sauce on the broccoli salad.