According to the Data Sheet, Pin E10 is automatically set for Function_0 (GPIO_SS), I would like to set it to Function_1 (PWM). Is there a way to do this utilizing registers and without re-configuring the firmware, or is there a good reference to configuring the firmware for the Quark?
In QMSI you can use qm_pmux_select() function. In Zephyr - pinmux_pin_set(). Note that Zephyr pre-configures some pin multiplexers in boards/<board_name>/pinmux.c.
I'd recommend against using registers directly for compatibility reasons. But if you really want to do that - check QMSI qm_pmux_select() implementation in qm_pinmux.c