Learn Portuguese (Portugal): Ele quase que estilhaçou o vidro incandescente! - He almost shattered the incandescent glass!

The word que here is an expletive particle (called a partícula de realce in Portuguese). It doesn't add any grammatical meaning and doesn't translate to anything in English. Its only job is to add a little emphasis or make the rhythm of the sentence flow better. You can absolutely drop it and say Ele quase estilhaçou, which means exactly the same thing.

We've learned that expressions ending in que often require the subjunctive. Why is estilhaçou in the indicative?

That is a great observation, and a very common trap! Because que is just an optional emphasis word here and not a conjunction linking two clauses, it doesn't trigger the subjunctive. The verb estilhaçar stays in the normal past tense (Pretérito Perfeito), exactly as it would if you simply used quase.

Could I use partiu or quebrou instead of estilhaçou?

You could, but estilhaçou is much more descriptive. While partir and quebrar just mean to break, estilhaçar specifically means to shatter or to smash into tiny pieces (estilhaços). It makes the sentence feel much more dramatic, which pairs perfectly with the emphatic quase que.

The adjective incandescente comes after vidro. Could it come before the noun for emphasis?