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1
Preheat the oven to 450 degrees and cover two baking sheets with foil (for easy cleanup). Slice the Brussels sprouts in half and pile them onto one of the baking sheets. Slice the acorn squash lengthways into thin slivers and put them on the other baking sheet.
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2
Pour olive oil over both trays of vegetables -- about two tablespoons on each tray. Toss with your hands to cover the vegetables with oil, and spread them out evenly. Turn the Brussels sprouts so that they're all cut-side down. Sprinkle salt over both trays, about half a teaspoon on each. Put both trays into the oven and roast until the outer leaves of the sprouts are very dark brown, and the underside of the squash is brown and crispy, about 20 minutes depending on your oven. Pour off excess oil immediately after removing the vegetables from the oven to prevent it being absorbed.
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3
Meanwhile, make the aioli. Mash the garlic clove with a pinch of kosher salt in a mortar and pestle. Put the remaining half cup of olive oil into a small jug or measuring cup with a spout. In a mixing bowl, whisk the garlic with the egg yolk and half a teaspoon of water. Using a hand or electric whisk, keep whisking vigorously while dribbling in the oil, drop by drop. As the oil is absorbed, the mixture will emulsify, becoming opaque and thick. Keep whisking and dribbling until all the oil is incorporated.
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4
In the mortar and pestle, mash together the anchovies and capers to a paste. Add the sriracha and lime juice, stir together, and whisk the mixture into the aioli. Taste and add salt if necessary, which it probably won't be.
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5
To serve, smear a couple of tablespoons of the aioli on a plate, then arrange the warm crispy sprouts and squash on top.
