How to run an electric motor at constant current?
The short answer- use current control. A longer answer- you have to learn how to drive a brushless motor, this is not trivial. 

The common system is like that:

MCU (firmware that runs position control on top of velocity control on top of current control) creates voltage (by changing PWM duty cycle on the three phases of the motor). The feedbacks are: current from two phases, position from encoder or hall effect sensors.

The voltage command from MCU must be calculated according to current loop requirements and motor position, to take into account the motor commutation.

I know, all that sounds gibberish, but it's only because the matter itself requires many many more details to know. In the end, driving a motor with constant current is really a trivial task.

Another thing i would suggest to think about, is what will happen with constant torque. I mean, at first the motor will accelerate, at the end, if you don't have load to compensate, the torque will be zero, because the voltage will be saturated and speed will be at it's maximum.