Combine 1 tablespoon tepid water and saffron in a small bowl; set aside. Bring a large pot of water to a boil. Add beets and cook until tender, about 20 minutes.

Meanwhile, fill a large bowl halfway with cold water and several ice cubes; add lemon juice.

Peel artichoke stems. Cut off top 1/2 inch of 1 artichoke with a serrated knife. Bend outer leaves backward until they snap off close to base, then discard several more layers of leaves in same manner until you reach pale yellow leaves with pale green tips. Cut off green tips. Trim dark green fibrous parts from base and sides of artichoke with a small sharp knife, then cut in half lengthwise and submerge in acidulated water. Repeat with remaining artichokes.

When beets are tender, drain and set aside. Drain and pat dry artichokes.

In a large skillet, heat oil over medium-high heat. Add artichokes; cook, stirring occasionally, 3 minutes. Add broth and gently simmer until artichokes are tender, 12 to 15 minutes. Add saffron mixture and cook 1 minute more. Drain artichokes, discarding cooking liquid.

In a medium saucepan, combine milk, cheese and salt. Over low heat, gently warm mixture, whisking occasionally, until smooth and warmed through, about 10 minutes. Divide sauce among four serving bowls. Top with beets and artichokes. Drizzle with oil. Serve warm.