Slice the tomatoes widthwise into 1/4" to 1/2" slices. I like to use a vegetable mandoline for ease and consistency, but a serrated knife also works well.
Lay the tomato slices onto a large cutting board and sprinkle evenly with the salt. Let the tomatoes sit until the salt draws out some of their water, about 10 minutes.
Meanwhile, prepare your batter station. Combine 1/2 cup oat flour and the arrowroot powder in a shallow bowl and whisk until combined. Place the remaining 1/2 cup oat flour, cornmeal, baking soda, pepper, paprika, and cayenne into a second shallow bowl and whisk until combined. Place the butter milk and egg into a third shallow bowl and whisk until smooth. Arrange the bowls so that the oat/arrowroot mixture is on the left, the egg/buttermilk mixture is in the center, and the cornmeal/oat/spice mixture is on the right.
Dab the tomato slices with a cotton kitchen towel (or paper towels) to wick away the moisture, being careful not to crush the tomatoes.
Dip a tomato slice into the oat/arrowroot mixture, then shake off the excess flour. Dip the tomato slice into the egg/buttermilk mixture, and then dip into the cornmeal/oat mixture. Set the slice aside onto a clean cutting board or plate and repeat with the remaining slices.
Heat half the oil in a 10" cast iron skillet over medium heat. Add a tiny sprinkle of cornmeal to test the heat of the oil - if it gently sautés, the oil temperature is perfect. If it burns, it's too hot. Adjust the heat as necessary, then add the battered tomato slices to the oil. Don't overcrowd the pan. Saute 2-3 minutes per side, until golden and crispy. Remove the fried green tomato slices with a spatula and drain on a towel lined plate. Continue to fry the tomatoes in batches, adding additional oil as needed to maintain a quarter inch depth in the skillet.
Make the aioli. Put the egg, garlic, lemon juice, and both olive oils into a glass or bowl large enough to fit the head of an immersion blender. I use a widemouth mason jar. Place the immersion blender into the jar and blend until the oil and egg are emulsified. As the aioli starts to come together, slowly lift and pull up on the head of the immersion blender so that the blade remains just above the surface of the sauce. Taste for seasoning and add salt and pepper as needed, then blend again.
Serve the fried green tomatoes with the dipping sauce. The tomatoes can be kept in the fridge for up to three days in a sealed container. Gently reheat in a cast iron skillet with a teaspoon of olive oil over medium heat. The aioli can be stored in the fridge in a lidded jar for about two weeks.