Of course you can read out the flash memory of an ESP microcontroller. This is also possible with the ESPEasyFlasher but it is not recommended. Because in this case you will get also memory parts that are not necessary. After this you have a big image that needs longer time to write it back to an ESP.

To get the information where the binaries will store by Arduino IDE and which esptool parameter will be used enable "Show verbose output druing: upload" in the Arduino IDE -> File -> Preferences, see screenshot below.

Now, you can upload the sketch. In the logging you will find the following output:

Sketch uses 324380 bytes (31%) of program storage space. Maximum is 1044464 bytes. Global variables use 28576 bytes (34%) of dynamic memory, leaving 53344 bytes for local variables. Maximum is 81920 bytes. C:\Users\andre\AppData\Local\Arduino15\packages\esp8266\tools\python3\3.7.2-post1/python3 C:\Users\andre\AppData\Local\Arduino15\packages\esp8266\hardware\esp8266\2.7.4/tools/upload.py --chip esp8266 --port COM3 --baud 921600 --before default_reset --after hard_reset write_flash 0x0 C:\Users\andre\AppData\Local\Temp\arduino_build_806841/ESP_jQuery_Mobile_Interface.ino.bin

This input you can use to create your eef files and you can find the binary file here in the example:

C:\Users\andre\AppData\Local\Temp\arduino_build_806841/ESP_jQuery_Mobile_Interface.ino.bin

This binary or in case it is more than one binary file. Put the binary files to the ESPEasyFlasher package, have a look to the examples.

If you have a file system you can use the same way, create the filesystem binary with the Arduino IDE and write it to the ESP. I am using the following tools for it ESP32 Sketch Data Upload and ESP8266 LittleFS Data Upload: