For the slow-roasted tomatoes, preheat the oven to 150C/300F/Gas 1.
Place the tomatoes, cut-side up, onto a roasting tray.
Sprinkle with the thyme, salt and freshly ground black pepper, to taste, and olive oil.
Place in the oven and cook for 1-2 hours until the moisture has mainly evaporated and the tomatoes have dried out.
Remove the tomatoes from the oven and cool.
For the béarnaise sauce, place the tarragon vinegar and diced shallot into a small saucepan and bring to the boil. Turn down the heat and simmer until the liquid has reduced by half.
Strain the liquid into a heatproof bowl and add the egg yolks.
Place the bowl over a pan of simmering water and whisk together until thickened and light in colour.
Gradually add the melted butter, whisking constantly. Season with salt and black pepper and add the chopped tarragon leaves.
Turn off the heat and leave the bowl over the warm pan until ready to serve.
For the rib-eye steak, season the steaks with salt and freshly ground black pepper.
Heat a frying pan until very hot and add the butter and olive oil.
When the butter is foaming, add the steaks and fry on each side for 2-3 minutes, depending how you like your steak cooked.
Remove from the pan and rest in a warm place for 2-3 minutes before serving.
For the fat chips, heat a deep-fat fryer to 140C/285F.
Slice the potatoes into 1.5cm/½in thick long chips.
Place in the fat fryer and cook for 5-6 minutes until just tender, but without colour.
Using a slotted spoon, remove the chips from the fryer and turn up the heat to 190C/375F.
Lower the chips back into the fat and cook for 3-4 minutes until golden-brown and cooked all the way through. Remove the chips again using a slotted spoon and drain on kitchen paper. Season with salt to taste.
To serve, pile the chips on one side of each plate. Place the steak next to the chips along with a spoonful of béarnaise sauce. Place three tomato halves on each plate and serve at once.
