Section 7.5.1 of the data sheet shows how to use two 8-bit exchanges to achieve a 16-bit exchange.
Don't forget that every SPI operation is an exchange. Therefore you need to start by writing a value to the SPI master that will contain the correct bits to request a read, , pick up the value just received, perform a 2nd exchange where you send anything (probably 0x00) and then merge the two received values together to get the 12-bit value.
You will need to select the SPI slave yourself before doing the above and then deselect it afterwards. (At least that is how I read the last paragraph of Page 135 of the Atmega16 data sheet.)
Susan