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Prick potatoes with a fork. Microwave, covered with a paper towel, on high power until tender, 10 to 12 minutes.
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Cutting lengthwise, remove top fourth of each potato and scoop out centers (reserve for another use), leaving 1/2-inch-thick walls.
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Whisk together mayonnaise, 1 Tbsp water, tomato paste, anchovy paste, and 11/2 Tbsp mustard.
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Preheat oven to 350°F with rack in middle.
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Cook onion in oil with a pinch of salt in a small heavy skillet over medium heat, stirring occasionally, until golden, about 10 minutes.
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Meanwhile, tear bread into small pieces, then soak in milk in a medium bowl until very soft. Mix in sausage, parsley, onion, and 2 Tbsp mayonnaise sauce with your hands.
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Divide filling between potatoes, then coat skins lightly with oil. Bake in a 4-sided sheet pan until sausage is cooked through, about 30 minutes.
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While potatoes bake, whisk 1 Tbsp mayonnaise sauce with lemon juice, remaining 1 1/2 tsp mustard and 1/2 tsp water, and salt and pepper to taste.
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Remove potatoes from oven and turn on broiler. Spoon remaining mayonnaise sauce over filling and broil 3 to 4 inches from heat until charred in spots, 3 to 5 minutes. Cool slightly.
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Toss greens with dressing and serve with potatoes.