16 Bit SPI mode is a special feature of the 16-Bit Microchip processors. Most micros don't have it. Im not sure, if you can probably access ADE7753 in 16 bit mode, but I suggest to use 8 bit mode as anyone does.
Actually, the ADE7753 accesses aren't generally 24 Bit. They are variable in length and made by sending one address byte and reading or writing one to three data bytes, depending on respective command. It's also explained in the datasheet in terms of byte accesses, as they are provided by the SPI standard and available with any SPI capable micro.
Of course you have to disassemble and assemble 16 and 24 bits to 2 and 3 bytes. A very basic programming task, that has several solutions in any programming language. I think, that Microchip also has programming examples for Analog energy chips or 24 bit ADC. CCC C at least has it.