Step 1Spray the bottom of a large skillet with nonstick spray, or use a nonstick skillet. Fill the skillet 3/4 full with water; bring to a boil.
Step 2Meanwhile, cook the shallot in 2 tablespoons of butter in a saucepan over medium-low heat until softened. Stir in the flour until blended and slightly cooked, 1 minute. Stir in the milk and cream. Heat to boiling, stirring constantly, then boil and stir 1 minute. Stir in the salt. Remove from the heat; strain the sauce through a fine mesh strainer and stir in the tarragon. Cover; set aside.
Step 3Heat the oven to 375 degrees.
Step 4Reduce the boiling water to a simmer. To poach the eggs, work with 2 at a time. Crack the first egg into a small bowl, then carefully pour into the water; repeat with the second egg. Cook until the whites are barely set, about 1 1/2 minutes, then remove them gently using a slotted spatula to a small buttered ramekin. Repeat with the remaining 2 eggs.
Step 5Divide the tarragon sauce between the ramekins, spooning over the eggs, then sprinkle with the cheese and bread crumbs. Bake until hot and bubbly, 5 to 10 minutes.