Heat the oven to 170C (160C fan)/350F/gas 4. Spread out the slices of bread on two baking trays and bake for 10 minutes, until lightly toasted and dried out, then leave to cool.
Grease a 30cm x 20cm and at least 6cm-deep baking dish with oil.
Put the kale, herbs, chilli, shallot and garlic in a food processor, pulse a couple times until finely chopped, then add 90ml oil, three-quarters of a teaspoon of salt and a good grind of pepper, and pulse to a coarse paste.
Measure out 75g of this pesto and set aside. Use the rest generously to coat both sides of each slice of bread. Layer the bread slices in the baking dish at an angle, so they overlap a little, and sprinkle the cheeses in between as you go.
In a medium bowl, whisk the eggs, yolks, milk, cream, mustard, three-quarters of a teaspoon of salt and a good grind of pepper. Pour over the bread, cover with a piece of greaseproof paper, then weigh down the strata with a heavy baking dish.
Refrigerate for at least four hours, and preferably overnight.
Take the strata out of the fridge an hour before baking, and remove the weight and greaseproof paper. Heat the oven to 190C (180C fan)/390F/gas 6.
Cover the strata tightly with foil, bake for 30 minutes, then remove the foil and bake for 25 minutes more, or until deeply golden and bubbling. Leave to cool for at least 15 minutes before serving.
Meanwhile, combine the reserved pesto with the capers, vinegar and remaining three tablespoons of oil. Serve the strata directly from its dish with the pesto alongside.