For the asparagus, preheat the oven to 425°F, with a rack in the top third.
On a sheet pan, toss the asparagus with the oil and salt and pepper to taste and roast until just tender, but not browned, about 10 minutes. Remove the pan from the oven and set it aside. Reduce the oven to 350°F.
For the bearnaise, in a small saucepan, combine the shallots, vinegar, wine, dried tarragon and a hefty pinch of salt. Bring the mixture to a boil, then reduce the heat to low and simmer 3 to 4 minutes until the liquid is reduced to about 1 tablespoon. Add the cream cheese, a tablespoon at a time, and whisk until it has been incorporated. Whisk in the milk, fresh tarragon and salt and pepper to taste and keep the sauce warm while you prepare the eggs.
For the eggs, in a deep, straight-sided skillet, add 2 inches of water and 1 tablespoon vinegar and 1 teaspoon salt for every 6 cups of water, then bring it to a boil. Break an egg into a fine-mesh strainer set over a bowl. Tilt the egg around in the strainer and bang the strainer a few times on top of the bowl to let all the loose egg white fall through the strainer. (That ensures that the poached egg has no straggly whites; if you don't mind straggly whites, skip this step.) Repeat with the remaining eggs, transferring the drained eggs to individual ramekins or small bowls. Discard the loose whites.
When the water is at a boil, turn off the burner, very gently slide the eggs into the water and cover the pan. Cook the eggs in the hot water for 2 to 3 minutes or until they have reached the desired degree of doneness (the residual heat will cook them). Transfer them with a slotted spoon to a kitchen towel to get rid of excess water before plating.
While the eggs are poaching, return the asparagus to the oven to reheat it.
To serve, place a piece of toast on each of 4 plates and top each piece with four stalks of asparagus and an egg. Spoon some of the bearnaise on top and serve right away.
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