Combine the shallots, vinegar, lemon juice and salt in a small bowl and toss to combine. Let sit for 15 minutes, tossing a few times to make sure the shallots are coated in vinegar and lemon juice mixture.
While the shallots are quick pickling, make the smoky white beans. Place two tablespoons of extra virgin olive oil and the garlic in a medium saucepan over medium heat. Cook, stirring occasionally, until the oil is hot and the garlic just starts to turn golden, about 1-2 minutes. Add the Pimentón de La Vera and continue to cook for another 30 seconds. Add the white beans, stir to coat in the pimentón garlic oil and continue to cook until the beans are heated through, about 2-3 minutes. Turn off the heat, add the remaining three tablespoons of extra virgin olive oil and gently mash the beans with the back of a spoon, leaving some of them whole (you want a rustic, yet spreadable texture). Season with salt to taste and set aside.
To finish the shallots, add the chopped herbs and toss to combine.
To assemble, slather each piece of bread with an even layer of the smoky white bean spread. Gently break the cod fillet into large chunks and divide among the four pieces of bread, piling it on top of the white beans. Garnish with the herby, quick-pickled shallots.