Step 1In a food processor or blender, combine the tomatoes, lemon juice, horseradish, shallots, Worcestershire sauce, red wine vinegar, black pepper and salt. Pulse or blend briefly until the mixture is combined but still a bit chunky.
Step 2Remove and place in a bowl. Fold in the green onions and adjust the seasoning.
Step 1In a small bowl, combine the lemon juice, vinegar and water. Set aside.
Step 2Finely chop the garlic and then smash with the side of a knife, smearing the garlic into the cutting board.
Step 3Place the egg yolks in a medium bowl. Place a damp towel on the counter in a ring; put the bowl in the ring and tighten the towel to keep the bowl from spinning as you add the oil.
Step 4With the bowl in place, whisk the egg yolks a bit. While whisking, add three drops of canola oil to the egg yolks. Once incorporated, slowly whisk in another three-fourths cup of the canola oil. The mixture will thicken a bit.
Step 5Whisk in one-fourth of the lemon juice, vinegar and water mixture. Alternating between the oils and the liquids, repeat this process until all the canola oil, olive oil and liquids have been added. The mixture should be smooth and creamy. Whisk in the garlic and salt. Cover and refrigerate.
Step 1Peel the celery root and cut it into 12 half-inch-thick wedges. Place in a small pot of lightly salted water and boil gently until knife-tender, about 3 to 4 minutes. Place on a towel to drain. Reserve the water to blanch the carrots and green beans.
Step 2Snip the tops of the carrots, keeping 1 inch of the stem intact. Rinse the carrots and rub with a clean kitchen towel to remove any dirt. Place in the pot of lightly salted water and boil gently until knife-tender, about 3 to 5 minutes. Place on a towel to drain. Reserve water for the green beans.
Step 3Clean and trim the green beans. Place in the water and cook until tender, about 3 to 4 minutes. Remove and submerge the beans in ice water until chilled. Place on a towel to drain.
Step 4Peel the onion. Slice into rings one-eighth inch thick.
Step 5Rinse and dry the acorn squash half. With the skin on, slice cross sections one-eighth inch thick and carefully remove seeds and pulp.
Step 6Remove and discard the mushroom stems. Slice the mushrooms into quarter-inch strips, being careful not to break the strips.
Step 7Pour the oil into a large pot. Clip a thermometer onto the pot if possible. Heat the oil to 360 to 375 degrees, then reduce the heat to low to maintain the temperature.
Step 8Pour the buttermilk into a large bowl deep enough for the buttermilk to cover the vegetables. Add the vegetables and allow to sit for a minute, stirring to ensure even coating. You may have to do this step and the cooking of the vegetables in batches.
Step 9In a separate bowl, combine the flour and semolina.
Step 10Lift the vegetables from the buttermilk, allowing them to drain through your fingers, then place in a large bowl. Take a handful of the flour mixture and dust and toss the vegetables until they are evenly coated. This may take a few dustings and a little skill, moving the bowl around at the same time as you are adding the flour mixture.
Step 11Place the coated vegetables into the hot oil, using tongs if desired. Cook just what will easily fit in the pan. When you first add the vegetables, the temperature will drop, so increase the heat a bit. Cook until tender, about 3 to 4 minutes. The onion takes less time, so you may want to fry that separately.
Step 12Remove the vegetables from the oil when they are golden brown and drain on paper towels. Season with salt. Repeat until all the vegetables have been fried. Place fried vegetables on a baking sheet in a 200-degree oven to keep them warm while you're frying.
Step 13Arrange the vegetables on a plate and serve with spiced tomato sauce and aioli for dipping.