Heat the olive oil in a deep skillet over medium heat until it shimmers. Add the shallots and thyme and cook until the shallots are browned on the edges. Fish out and discard the thyme. Add the cauliflower and stir to coat with oil. Increase the heat to medium-high, pour in the beer, and bring to a boil. Reduce the heat, cover, and steam the cauliflower until fork-tender, about 7 minutes. Uncover and cook until all the liquid has evaporated, another 5 minutes. Remove the pan from the heat and stir in the parsley, salt, and pepper. Cool to room temperature.
Fill the bottom crust with the cauliflower filling. Whisk together the milk, eggs, Worcestershire, and mustard in a small bowl until the eggs are thoroughly incorporated. Pour over the cauliflower filling, sprinkle with the cheddar cheese, and cover with the top crust. Crimp and slash.
Combine the egg yolk and water and use a pastry brush to paint it on the top crust from edge to edge. Sprinkle with the Parmigiano and slip the pie into the hot oven (on top of the steel, stone, or baking sheet if using). Bake for 20 minutes. Reduce the temperature to 375˚F and bake until the crust is bronzed and the filling is bubbling, 25-40 minutes. Cool slightly before slicing into portions.