Learn Hardcore Japanese: ryourinin ga tatemono no minamigawa de yasai wo yaite iruno ga mieru. - I can see the chef grilling vegetables on the south side of the building.

The first が (in 料理人が) marks the subject of the subordinate clause “料理人が … 焼いている.” The second が (in のが見える) marks the subject—or more precisely, the “thing that is visible”—for the main verb 見える. Grammatically, the entire nominalized clause “料理人が建物の南側で野菜を焼いているの” is what you “see,” so it takes が as the marker of what is visible.

What is the function of の after 焼いている?

Here, の is a nominalizer. It turns the verb phrase 焼いている (“is grilling”) into a noun phrase (“the act of grilling”). That allows the whole clause to be the subject of 見える:
料理人が建物の南側で野菜を焼いているのが見える
(“You can see the [act of the cook grilling vegetables on the south side of the building].”)

Why is 焼いている used instead of 焼く?

焼いている is the progressive form, showing that the action is ongoing—“is grilling.” If you used the plain form 焼く, it would simply state a habitual or general fact (“grills vegetables”) rather than what’s happening right now.

Why is で used with 南側 rather than に?

で marks the location where an action takes place. You grill で that spot. By contrast, に would mark a destination or point of arrival (e.g. 「南側に行く」 to go to the south side). So 南側で tells us “at/on the south side.”

What is the difference between 建物の南側 and 南側の建物?

建物の南側 literally means “the south side of the building.” The focus is on a part of the building.
南側の建物 would mean “the building on the south side [of something else],” so here “building” is modified as being located on a south side relative to some context.

Why is 野菜 marked with を?

野菜を marks 野菜 as the direct object of the verb 焼いている. In Japanese, を always flags what the verb acts upon—in this case, the cook is grilling the vegetables.

What’s the nuance between 見える and 見られる?

見える means “(it) is visible” or “(I/one) can see (it),” focusing on something being perceptually apparent without emphasizing ability or permission.
見られる is the potential form of 見る (“to see”), meaning “can see” in the sense of having the ability or opportunity, and sometimes “be seen” if treated as passive.
In our sentence, 見える simply reports that the grilling is in plain sight.

Why is there no explicit subject (like 私 or あなた) for 見える?

In Japanese, subjects that are obvious from context are often omitted. With verbs like 見える, it’s understood that the speaker or a generic observer is doing the seeing. Adding 私が見える would sound redundant or awkward because 見える already implies who is perceiving.