DC voltage controlled current source circuit for high currents
Such high currents are best handled efficiently.  Getting rid of the heat from inefficiency will take more trouble, expense, and space than designing a better circuit in the first place.

My first reaction is to have the microcontroller control a low side switch with PWM.  The thermal time constant of even a thin wire will be long compared to what modest PWM hardware in a micro can do.

The nichrome wire won't get hurt by pulses, but you can put a inductor in series with it if you really want to filter the current for some reason.  Since the wire looks like a resistor electrically, all you need is a power supply, the wire, and a switch in series.

Added in response to schematic

I wouldn't try to invent your own FET gate driver.  You want the edges to be nice and crisp.  8 A is a bit too much to reasonably find a FET that can be driven directly from a digital signal.  With 8 A thru 1.5 &Omega;, you obviously have a 12 V power supply available.  That would be ideal for powering a FET driver.  The switching edges should be nice and sharp, and the result quite efficient.  No heat sink should be needed.

For example, let's say you get a FET with maximum Rdson of 15 m&Omega;.  That would only dissipate just about 1 W.  Something like a DPAK with the right copper pads can dissipate that directly.  Since nearly 100 W would be going into the wire, the overall setup is quite efficient.  And, at these low voltages, 15 m&Omega; is no stretch at all.

Note that in this case, the PWM duty cycle controls the average power, not the average voltage, as it would with a inductor and diode in there.  However, that doesn't make it harder to control.  If this is part of a larger control loop, linearly controlling the power is probably better than controlling the square of the power, which is what voltage control would be.

Supply voltage

I just noticed due to a comment from Trevor that you plan to use a 24 V supply.  Don't do that.  Use a supply that is the voltage the wire needs at full on.  A 24 V supply will cause twice the dissipation in the same FET as a 12 V supply when using the purely resistance-switching scheme I'm talking about.  Fortunately, 12 V is common DC power supply value.  Then you can also run the FET driver directly from the same supply.  24 V would be too high for that.