Step 1Heat the oven to 400 degrees. Hold a lobster securely against the cutting board, wrapping your hand in a kitchen towel to protect you from the spines. Kill the lobster by piercing the head well behind the eyes with a heavy chef's knife and then slicing down, essentially cutting the head in half lengthwise. Set the lobster aside for a minute until it is still. Repeat with the other lobster.
Step 2Meanwhile, make the aioli: Place the garlic in a large, heavy mortar along with the salt and pound with a pestle into a smooth, sticky paste. Using the pestle to stir, beat in the egg yolks and stir until the mixture is smooth and lemon-colored, about 30 seconds. (Alternatively, the mayonnaise can be made with a whisk. In a medium bowl, mash the garlic using the back of a large spoon. Whisk in the egg yolks, stirring until the mixture is smooth and lemon-colored, about 30 seconds.)
Step 3Switch to a whisk and stirring constantly, begin to beat in the olive oil, adding just a drop at a time, until the oil begins to emulsify with the egg yolks. Do not attempt to rush the process by adding the oil too fast. After you've added about one-fourth cup this way, you can increase the flow to a thin stream. If at any time you see oil beginning to gather separately from the yolks, immediately stop adding oil while continuing to stir vigorously. Very shortly the mixture should come back together.
Step 4When you have added about one-half cup of olive oil, and the aioli is quite thick, stir in the saffron threads and the hot water, which will loosen it considerably. Then begin whisking in the vegetable oil, starting again with a drop at a time.
Step 5When you've worked in about one-fourth cup of vegetable oil, whisk in the lemon juice and taste the sauce. The texture should be creamy, not stiff and sticky. The flavor should be very garlicky but sweet, with a balance of garlic, olive oil, lemon and saffron flavors. If it's still too garlicky, stir in a little more oil. Add more salt and lemon juice if necessary. (If the sauce breaks -- the oil and yolks separate and won't come back together -- it can be easily fixed. Add a whole egg to a blender and puree it until smooth. Pour the broken aioli mixture into a measuring cup and, with the blender running, slowly add it to the blender. When it is completely incorporated, slowly add more oil with the blender running until you have the texture and flavor you prefer.)
Step 6Cover tightly and refrigerate until ready to use, but allow it to return to room temperature before serving. Aioli should be made no more than a couple of hours in advance.
Step 7Place the salt in a large bowl and stir in 1 1/3 cups water until the texture is that of gritty snow. Line a jellyroll pan with parchment paper and spread some of the salt in a layer about one-fourth-inch deep that is roughly the size of the lobsters. Place the lobsters on top of that, stretching them out lengthwise, and trimming the long antennae. Mound the rest of the salt on top, covering the lobsters completely, and smooth the surface. Roast the salt-encased lobsters in the oven for 30 minutes.
Step 8After 30 minutes, remove the lobsters from the oven and let stand 5 more minutes to finish cooking. With a sturdy metal spoon or chef's knife, chip a crack around the base of the salt crust and carefully lift off the top.
Step 9Transfer the lobsters from the salt to a cutting board. Separate the tail from the head. (Flex the tail and you'll see a good place to cut just behind the carapace.) Use a sturdy pair of kitchen scissors to cut a slit through the shell the length of the tail. Using a kitchen towel to protect your hands, pop the shell in half and remove the tail meat. Carve the tail into medallions and arrange them on a tray with a bowl of saffron aioli.