Article 2 Section 1

The Electors shall meet in their respective States, and vote by Ballot for two Persons, of whom one at least shall not be an Inhabitant of the same State with themselves. And they shall make a List of all the Persons voted for, and of the Number of Votes for each; which List they shall sign and certify, and transmit sealed to the Seat of the Government of the United States, directed to the President of the Senate. The President of the Senate shall, in the Presence of the Senate and House of Representatives, open all the Certificates, and the Votes shall then be counted.

There is nothing in that text to suggest that the Legislative Branch has any authority to do anything, save be present as witness to the counting.

The count is entirely under the control of the president of the Senate, the vice president of the nation.