Do gaps like $\infty$ & on have birthdays?
I haven't ever seen anywhere that they do, & I think the answer is that they don't (although I'm not sure how to prove it).
This got me wondering, if they don't have birthdays, when do they become accessible?
The technically-accurate answer is this:
There are a few different ways to define a birthday, but they are all an ordinal. No ordinal corresponds to a proper class, so a gap has no "formal birthday" by definition. (A gap like $\{\text{[class of all negative surreals]}\mid\}$ might have birthday $0$ since it should equal $0$, but no formal birthday since the left class isn't a set.) The gaps aren't part of the recursive construction of the surreals, so they can't get a birthday from the construction.
I want to emphasize that birthdays are a sort of metaphor due to the inductive definitions of the surreals. In real time, if I write down the definition of what it means to be a surreal, then I gain access to the proper class of all surreals today.
I'll write more below, but I'd say a question like this is best answered by studying set theory, and then your understanding would answer this and a million more related questions you might have.
Well, at any ordinal birthday, no proper class of surreals has been created yet. So if you force me invent a new metaphor to answer "when", I'd have to say "after all the birthdays".
And if you force me to give a name to the "day" or "days" that would be appropriate, maybe "the class of all ordinals" would be a candidate name, simply because it contains all the ordinals that came "before", just like a limit ordinal would.
As an aside, note that the relationship between possible sizes of the left and right classes and the birthday could fail for the gaps for set theory reasons. Without something like the axiom of limitation of size, a proper class could be bigger than the class of all ordinals, as discussed in this answer of Noah Schweber.