Place the egg whites and sugar in a spotlessly clean heatproof bowl (the bowl of your stand mixer if you have one) and place the bowl over a pan of gently simmering water, make sure that the base of the bowl doesn't touch the water.
Gently whisk the eggs (with a hand whisk, not an electric one) until the sugar has dissolved, if you rub the mixture between your fingers it should feel smooth with no hint of graininess. It should measure 60C/140F on a thermometer.
Remove the bowl from the heat and whisk with an electric mixer/stand mixer on high until the egg whites have formed a stiff, glossy meringue and the mixture is completely cold, the base of the bowl should feel cool to the touch, this will take about 10 minutes.
Turn the mixer down to low and gradually add the butter, a tablespoon at a time, mixing to incorporate after each addition. The mixture may curdle or begin to look soupy, just keep on whisking and it will come back together.
Once all of the butter has been incorporated and the buttercream is smooth and silky, whisk in the blackberry puree gradually, a spoonful at a time. The mixture shouldn't split, but if it does don't worry! Just place a couple of heaped tablespoons of the buttercream in a small bowl and microwave it until it has melted, then drizzle it back into the rest of the buttercream slowly while whisking on a high speed, after a couple of minutes of whisking it will come back together.
Finally whisk in the vanilla extract and add the food colouring a little at a time until you reach your desired shade.