Adjust an oven rack to the lowest position and preheat the oven to 325°F / 165°C. Lightly coat 6 (6-ounce) ramekins with cooking spray and arrange them on a baking sheet.
Using a rolling pin, flatten each slice of bread to about ⅛-inch thickness. Gently press one bread slice into each ramekin, lining the bottom and sides. The bread may overlap and extend over the top.
In a small bowl, combine the chopped mushrooms, sliced green onion, and tarragon.
Season with a pinch of salt and pepper.
Divide this mixture evenly among the bread-lined ramekins. Carefully crack one egg into each nest.
Top each with a sprinkle of Swiss cheese, paprika, and another light seasoning of salt and pepper.
Bake for 20 to 25 minutes, rotating the baking sheet halfway through. The nests are ready when the egg whites are fully set and the yolks are thickened but still creamy. For firmer yolks, bake for a few minutes longer.
Let the nests cool in the ramekins for 2 minutes before carefully running a knife around the edge to unmold.
Transfer to plates and garnish with fresh tarragon leaves, if desired.
