For the brioche, put the yeast in a small bowl. Put the milk in a small saucepan and warm through. Pour the milk over the yeast and mix. Leave in a warm place for 10 minutes, or until a light foam forms on the surface.
Put the flour, caster sugar and salt in a food processor with a dough hook attachment. Start the mixer and pour in the yeast mixture and the beaten eggs gradually. Knead for 5 minutes in the food processor.
Gradually add the softened butter, a teaspoon at a time, kneading well between each addition. It will take about 10 minutes to incorporate the butter and the dough will be very sticky.
Sprinkle the dough with flour and continue to knead lightly for 10 minutes or until the dough is very pliable, smooth and no longer sticky. Put it in a large greased bowl and cover with a clean cloth. Leave in a warm place for 1½ hours until well risen and spongy.
Preheat the oven to 180C/160C Fan/Gas 4.
Turn the dough out onto a lightly floured surface and knock it back with your knuckles. Shape into 6 round brioche rolls and leave in a warm place to prove for 45 minutes, or until the dough is well risen.
Meanwhile, make the lamb burgers. Put the mince in a large bowl and season with salt and pepper. Use your hands to shape the mince into 4 burger-shaped patties. Wrap the patties in cling film and put in the fridge for an hour.
Once the brioche dough has proved, brush the brioche rolls lightly but generously with the beaten egg yolk. Bake for 15-20 minutes until risen and golden-brown.
Meanwhile make the mint mayonnaise. Put the egg yolks, white wine vinegar and mustard in a food processor and blend until pale and creamy. With the motor running, pour in the oil in a very thin, steady stream until the mayonnaise thickens. Add the mint, mix and set aside.
For the tomato chutney, heat the oil in a small pan. Add the shallot, tomatoes and garlic. Cook for 5-8 minutes until softened. Stir in the vinegar and sugar and cook for 3-5 minutes, or until the chutney has caramelised slightly.
To cook the lamb burgers, heat a large frying pan and add the oil. Once hot, add the burgers and cook each side for 2-3 minutes, or until cooked through. Set aside to rest in a warm place.
For the homemade chips, heat the oil in a deep fat fryer to 180C. Alternatively, heat the oil in a deep, heavy-based frying pan until a breadcrumb sizzles and turns brown when dropped into it. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
Carefully put the chips in the deep fat fryer and cook for 3-4 minutes, or until golden-brown and crisp. Remove and drain on kitchen paper. Season with salt.
Halve the brioche buns and lightly toast. Spoon the tomato chutney on the base of the bun and top with the burger, gherkins, tomato slices, onion, lettuce, mayonnaise and place the other half on top. Secure with a cocktail stick if needed. Serve the burger with the fries alongside.