Tomato Carpaccio

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2 extra-large egg yolks, at room temperature

2 teaspoons Dijon mustard, at room temperature 

1 tablespoon chopped garlic (3 cloves) 

2 anchovy fillets, drained 

1/2 cup freshly squeezed lemon juice, at room temperature (2 to 3 lemons) 

Kosher salt and freshly ground black pepper 

3/4 cup good olive oil 

1/4 cup canola oil 

1/2 cup freshly grated Italian Parmesan cheese 

4 to 5 large, firm, ripe red summer tomatoes, preferably heirloom 

1/2 teaspoon fleur de sel 

2 tablespoons capers, drained 

1/4 cup julienned fresh basil leaves 

2-ounce chunk of Italian Parmesan cheese, shaved with a vegetable peeler