If using fresh artichokes, squeeze juice from lemon halves into a large bowl of cold water; add squeezed lemon halves to water. (Alternatively, if using frozen artichokes, omit lemon and water, and proceed to step 2.) Trim stem of 1 fresh artichoke to 1 inch in length, and snap off the tough outer leaves near the base. Continue snapping off leaves until only the central cone of yellow leaves with pale green tips remains. Cut off and discard top third of artichoke; retain the remaining artichoke heart. Trim away any remaining tough dark-green areas around the base using a small paring knife. Cut artichoke heart in half lengthwise. Scrape out and discard the hairy choke from the center of each half using a small spoon. Cut each half lengthwise into 4 even slices; place slices in bowl of lemon water. Repeat process with remaining artichokes. Drain artichokes well, and discard lemon halves. Set artichokes aside.

Heat 2 tablespoons oil in a large nonreactive saucepan over medium. Add shallots and salt; cook, stirring often, until shallots are softened but not browned, about 3 minutes. Add artichokes (fresh or thawed frozen) and remaining 2 tablespoons oil to pan; cook until slightly softened, about 2 minutes. Add wine, and bring to a simmer over medium-high; simmer, stirring occasionally, until mostly evaporated, 6 to 8 minutes. Add chicken stock; return to a simmer over medium-high. Reduce heat to medium-low; cover and simmer soup 20 minutes.

Remove soup from heat, and transfer to a blender. Secure lid on blender; remove center piece to allow steam to escape, and place a towel over opening. Process until smooth, about 1 minute. (Alternatively, use an immersion blender to puree.) Pour soup through a fine wire-mesh strainer set over a large heatproof bowl. Press mixture using the back of a spoon to extract all of the puree; discard fibrous solids.

Return soup to saucepan; heat over medium to warm through. (If a thicker soup is desired, simmer, stirring often, until slightly thickened, about 5 minutes.) Divide soup among 4 serving bowls; drizzle with extra-virgin olive oil, and garnish with cheese. Serve soup immediately with brioche toasts, if desired.