And back at the state level of politics

Not a great surprise, since we knew they weren't fans, but the Pioneer Institute is calling for Commissioner Chester to be fired.
UPDATE: I posted this over on Twitter:

Yes, @MassGovernor appts Sec'ty & selects chair from BoE members, but Sec'ty doesn't (correct me if I'm wrong) control DESE budget. 2/2
— Tracy Novick (@TracyNovick) January 15, 2015

Pioneer got back to me with this:

@TracyNovick It's in MGL, part of 2007-08 Patrick reorg, DESE firmly under exec branch, days of independent DESE gone
— Pioneer Institute (@PioneerBoston) January 15, 2015

...which is a little nonspecific. So I looked it up! The powers and duties of the secretary and the commissioner are spelled out in Chapter 69, sec. 1A, which says this about the budget:

The commissioner shall propose a budget to the board; said budget shall reflect the goals and objectives of the board and the secretary of education, in this chapter called the secretary. The board shall review and make recommendations regarding the budget to the secretary. The secretary shall then prepare and submit a budget request on behalf of the department to the house and senate committees on ways and means, the joint committee on education, and the secretary of administration and finance.

The Secretary thus fulfills his role as the connection between DESE and the Executive branch (which then proposes the budget to the Legislature). From this, it doesn't look as though the Secretary has actual veto power, though to interpret this generously, he presumably could refuse to forward the proposed budget to the Governor.
...which could get interesting...