Use Micro-Controller Instead of ESC
It sounds like what you're wanting to do is to roll your own onboard ESC as part of your quadcopter design to save weight instead of using a prepackaged ESC on its own PCB. That's a perfectly reasonable thing to do, but there's a little more involved than just the microcontroller. 

Most importantly, you need a power driver for each motor, either a dedicated BLDC driver IC or 6 discrete MOSFETs. If you use a microcontroller with enough PWM outputs, you won't need a separate BLDC controller IC in addition to the driver. Off the top of my head, there's STM32F0, PIC24, dsPIC, and LPC4350 chips that have dedicated precision motor control PWM timers.

I can't see a datasheet/wiring diagram for the motor you linked, but it has 3 wires (obnoxiously, all black) coming off of it, which means it's likely a standard 3-phase BLDC with no hall effect sensors built in. 

The term you want to Google is "sensorless BLDC control", which will come up with a whole bunch of application note PDFs from various microcontroller vendors. This method of control requires use 3 channels (per motor) of your microcontroller's ADC to measure back EMF in the motor in order to tell when to energize the next phase of the motor.