Learn Spanish (Latin America): Vamos a ver más burbujas a medida que la masa fermente. - We are going to see more bubbles as the dough ferments.

Because the action of the dough fermenting is an anticipated future event. The phrase a medida que (as) takes the subjunctive when describing a parallel progression that has not finished happening yet. If we were describing a habitual, experienced fact, like 'Every time we bake, we see bubbles as it ferments', we would use the indicative fermenta.

Could I use mientras instead of a medida que here?

Yes, but there is a slight difference in nuance. Mientras just means 'while', focusing on the time overlapping. A medida que emphasizes the parallel progression, meaning that the amount of bubbles we see will increase proportionally alongside the progression of the fermentation.

Is fermente an irregular verb?

It is a completely regular present subjunctive. The infinitive is fermentar (an -ar verb). To form the present subjunctive, you take the 'yo' form of the present indicative (fermento), drop the -o, and add the opposite ending, which is -e for the third-person singular form: fermente.

Does masa always mean 'dough'?

Not always. While it means 'dough' or 'batter' in a baking context, masa also translates to 'mass' in physics or general contexts, like la masa muscular for muscle mass. It can also mean 'the masses' when referring to a large crowd of people.