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deborahjarman
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Preheat oven to 175˚F (convection), or as low as your oven goes (200˚F will work).
Place the sliced tomatoes on a parchment paper-lined sheet pan. Bake for 2-3 hours, until the edges are mostly dried out. The center may still be a little soft – that's fine! (At 175˚F convection, this took us right around 2 ½ hours, but it may vary based on your oven.)
Remove from oven and increase heat to 400˚F.
On a nonstick sheet pan, portion out the parmesan into ~½ tablespoon piles, spaced about 2 inches apart. Place a dried tomato on each pile. Cover the sheet pan with aluminum foil and return to the oven for 8-12 minutes, until the parmesan is lightly browned around the edges. Remove from the oven and let cool for a minute before removing with a spatula. Let them cool completely before enjoying!
To make the aioli, add the egg, basil, garlic, dijon, lemon juice and salt to a vessel just slightly larger than your immersion blender. Blend together until smooth, then with your blender off but still at the bottom of the vessel, pour in the oil VERY slowly. Blend from the bottom and slowly lift the immersion blender upward until the aioli is emulsified (this shouldn't take long!)