Preheat the oven to 325 degrees F.
In a deep ovenproof pan, combine 1/2 cup olive oil, lemon juice, garlic, thyme, bay leaf, salt, and pepper. Add the frozen artichokes and mix to coat. Bring to a boil over medium heat, stirring occasionally. Cover with aluminum foil, transfer to the oven, and cook until the artichokes are browned in spots and tender when pierced, about 35 minutes. Allow the artichokes to cool in the liquid.
Remove the bay leaf. Put the cooled artichokes with the liquid into a blender. Add the remaining 1/4 cup olive oil and basil leaves and process until smooth.