hi,
If the MCU and EEP are in sockets, remove the two devices from the sockets.
Disconnect the scope from the pcb.
With a DVM on 'resistance' measure from the 0v pins on the sockets to the power supply ground [0v].
From your description , a ground connection is missing, so when you connect the scope, you are making the ground [0v] line thru the scope and power supply.
If you cannot unplug the MCU and EEP then, disconnect the scope, power up your pcb, measure from the psu 0v to the 0v pins on theMCU and EEP, also all the pins that are supposed to be connected to 0v.
If the DVM shows a voltage, that pin is not grounded.
Using a DVM on volts, I think you will discover one the 0v pins is 'floating'
EricG