In a medium bowl, combine the calamari bodies and tentacles with the milk and refrigerate overnight. In a small bowl, whisk 1 cup olive oil with the pimentón and refrigerate overnight. The next day, drain the calamari, pat dry with paper towels, and refrigerate until ready to use. Remove the pimentón oil 1 hour before serving.
In a small saucepan, cover the lemons with cold water, and bring to a boil. Cook the lemons for 40 minutes, and then drain and cover again with fresh water. Repeat this process 4 more times, and then drain the lemons and let stand until cool enough to handle. Halve the lemons and remove and discard the flesh from the peel. Heat a 9-inch cast iron pan over medium-high. Drizzle the lemon peel with 1 teaspoon olive oil and add to the skillet. Cook the peels, tuning once, until slightly charred, about 4 minutes. Transfer the peels to a plate and let the pan cool.
In a medium saucepan, cover the potatoes, garlic, thyme, and bay leaf with cold water and season generously with salt. Bring to a boil over high heat, then reduce the heat to medium and simmer until the potatoes are tender, about 15 minutes. Drain the potatoes, and discard the garlic and herbs. Let the potatoes cool for 5 minutes, then smash with the palm of your hand. Heat the butter in the cast iron pan over medium. Add the potatoes, and cook on one side until crisp, about 5 minutes. Transfer the potatoes to a plate.
Wipe the pan clean, then heat over high. Add the tomatoes and cook, undisturbed, until blackened, about 5 minutes. Transfer to the plate with the potatoes. Return the pan to high heat, add the bacon, and cook, turning once, until it begins to caramelize, 2 to 3 minutes. Push the bacon to the perimeter of the pan and allow some of the fat to burn off until the pan is almost dry, about 5 minutes. Working in batches, add the calamari bodies and tentacles, and cook, turning once, until charred on both sides, 2 to 3 minutes. Using a slotted spoon, transfer the calamari to a plate, drizzle with 1 tablespoon lemon juice, and season with salt and pepper. Remove the pan from the heat and let cool with the bacon.
Heat the broiler. Tear the brioche into 6 large chunks and place on a baking sheet. Cook the brioche, turning, until toasted all over, about 2 minutes. In a bowl, toss the avocado slices with the remaining 5 teaspoons olive oil and 1 tablespoon lemon juice, and the cilantro leaves. Season the avocado with salt and pepper.
To serve, divide the calamari, lemon peels, brioche, potatoes, tomatoes, and avocado among 4 plates, and then drizzle with the pimentón oil.