In a medium saucepan over medium-high heat, bring the butter, salt and 1 cup water to a boil. Reduce the heat to low. Add the flour and stir for about 1 minute. (The mixture will immediately form a sticky ball.)
Transfer the mixture to a stand mixer fitted with a paddle attachment and mix on medium-low until there is no longer steam. Lower the mixer speed to low, and add the eggs one at a time. The dough will be done once it clings to the paddle in "spokes" as it spins in the bowl.
Fold in the manchego and transfer to a pastry bag fitted with a large round tip. Dab the dough into 4 corners on a baking sheet and place parchment paper on top to stick it in place. Pipe 1 1/2-inch-wide rounds on the prepared baking sheet 2-inches apart, then sprinkle a little manchego on top of each. Bake for 15 minutes, then rotate the pan and lower the oven temperature to 375 degrees F and bake until deep golden brown, an additional 15 minutes. Cool to room temperature.
Meanwhile, combine the mayo and chipotle and set aside. Cut the Muenster to fit the sliders with a 2 1/2-inch round cutter. Divide the chorizo and form into 12 patties, then cook on a griddle or in a medium skillet in 2 batches over medium-high heat for 2 to 3 minutes. Flip and cook another minute, then top each with a round of the Muenster and continue cooking until the cheese is melted.
Cook the quail eggs in a buttered nonstick skillet over medium-high heat until sunny-side up, 1 to 2 minutes.
To assemble, split the gougeres. Top each of the bottoms with a chorizo patty, followed by an egg, followed by a pinch of micro greens. Smear a little chipotle aioli on the top halves and place on place on top of the sliders. Serve immediately.