Place a 9-inch cast-iron skillet over medium heat and sprinkle in the sugar. Allow the sugar to melt until caramelized smelling, gently stirring occasionally, 2 to 3 minutes. Add in the butter and whisk to combine. Standing back from the stove, whisk in the sherry vinegar. If any clumps of sugar occur during this process, don’t worry, they will melt in later.
Carefully add the shallots and pears to the pan, season with salt and pepper, turn the heat down to low and cook until slightly softened and caramelized, 4 to 5 minutes, gently stirring occasionally. Meanwhile, retrieve puff pastry from the fridge, unroll it and poke a few times with a fork to allow steam to escape.
Remove the pan from the heat, arrange the shallots and pears into an even layer and carefully drape the pastry overtop. Using a small spoon or butter knife, tuck the pastry down around the shallots and pears into the sides of the pan.
Transfer to the oven and bake until the pastry is golden, puffed and crisp, 35 to 40 minutes. Allow the tart to cool in the pan for 15 minutes before running a knife around the edge and, wearing oven mitts, very carefully and quickly inverting onto a rimmed serving plate, being careful of the hot caramel. Set aside.
For the garnish: In a large bowl, whisk together the mustard, sherry vinegar, honey and olive oil. Season with salt and pepper and toss in the arugula.
Top the tarte with the dressed arugula and scatter with the walnuts and blue cheese. Slice and serve.
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