I believe the Orthodox do perform confirmation at the same time as baptism.
The second rite of Christian initiation (or third for those enrolled as catcheumens prior to baptism) is understood in East and West alike as confiring the gift of the Holy Spirit (note: singular gift, not plural 'gifts'). It also in East and West alike serves as a sign of unity with the bishop, in whom the fullness of that gift is realized in terms of the order of the Church (even if not in his person).
In the West the rite must be carried out by the bishop himself, as it is performed by the laying on of hands by the bishop. It was thus, even in ancient times, separated from baptism in time for all save those baptized in the bishop's presence. This rite came to be called "Confirmation" as it was understood that the bishop confirmed what had been done at baptism.
It is my understanding that the long separation, and the requirement for "understanding" (thus classes) was an innovation in response to the protestants. It also, bizarrely, places this rite after the reception of the Eucharist ("First Communion"). To an Orthodox this is very bizarre: that one who has not receive the Holy Spirit should receive the Body and Blood of Christ.
In the East the Mystery is called "Chrismation" and the sign of unity with the bishop is the requirement that the oil used in the anointing have been blessed by the primate of the local church (whether he is titled Patriarch, Metropolitan or Archbishop). The newly baptized is immediately Christmated, and receives the Holy Mysteries of the Body and Blood of Christ at the first Divine Liturgy afterward (usually the same day or next day). Baptisms are usually done at the end of Orthros, just before Divine Liturgy in Greek usage or at the end of the "All-Night Vigil" the evening before in Slavic usage.
Chrismation is also often used for the reception of converts who have been baptized in the name of the Father and of the Son and of the Holy Spirit in a confession outside the Holy Orthodox Church which nontheless professes an orthodox doctrine of the Holy Trinity. (Though some may be received by profession of faith alone--Copts or Armenians for instance, and some in cases where it is unclear whether a proper baptism was done--e.g. Anglican, who nowdays might have been baptized in the name of the Creator, Sanctificer and Redeemer (Sabellius, call your office)--may be received by Baptism and Chrismation.)