In a saucepan, saute shallots in butter until tender. Add mushrooms; cook and stir until mushrooms release their liquid. Stir in flour until smooth. Gradually stir in milk. Bring to a boil; cook and stir for 2 minutes or until thickened. Add tarragon. Reduce heat; cook and stir for 5 minutes. Add cheese; stir to melt. Remove from the heat; keep warm.
In a small saucepan, warm the artichoke bottoms in water to cover. Season lemon juice and salt. Keep warm.
To poach eggs, place 2-3 in. of water in a large skillet with high sides; add vinegar. Bring to a boil; reduce heat and simmer gently. Break cold eggs, one at a time, into a custard cup or saucer; holding the cup close to the surface of the water, slip each egg into water. Cook, uncovered, until whites are completely set and yolks begin to thicken, about 4 minutes. With a slotted spoon, lift each egg out of the water.
To serve, top each muffin half with an artichoke bottom, poached egg and about 1/3 cup sauce. Garnish with a sprig of tarragon if desired.