You can do this a day ahead. Clean mushrooms, slice woody part of stem off. Halve the mushrooms and place in food processor. Rough cut the shallot and place in processor. Pulse until you have a fine dice on everything. Don't turn them to mush!
In sautee pan over medium heat, add butter, when foaming add mushroom mixture and cook 4-5 min, stirring frequently. Gradually season salt & pepper all along the way. Halfway through, add fresh thyme.
When mushrooms have lost their moisture, add white wine and reduce fully.
Remove from heat and refrigerate if using for later.
Build Wellington
Remove pastry dough just before cutting. You want it to stay firm. On a silicone mat or lightly floured surface, unroll sheet of dough. Cut a round disk, 7-8" diameter, using a biscuit ring or a knife. Layer it in the mini pie tins. There should be 1/2" of dough draped over the edges.
Lay a sheet of prosciutto in the very bottom of the pastry dough, folding it over so you have a double layer. Spread just under 2 tablespoons of duxelles over the meat, like you would spread jam on toast. Place another layer of prosciutto on top of the duxelles.
With your fingers, make a slight indent in the center of the meat. Enough to allow the yolk to sit in the center.
In a ramekin, crack an egg into it, careful not to break the yolk. Carefully pour the intact egg onto the the prosciutto ensuring the yolk sits in the center.
*If you skip the lattice, fold the excess edges of the dough over themselves so you make a nice pinched rim. With a pastry brush, baste edges of dough with egg wash.
Lattice pastry top
This step is optional. Cut your lattice from the remaining pastry dough using a lattice cutter. Stretch it out fully and cut a slightly smaller disk: 6-7". carefully place the lattice over the egg so the edges of the lattice touch the edge of the dough in the pie tin. Fold the excess dough edge over the edges of the lattice and pinch down to make a nice rim. With a pastry brush, baste edges of dough with egg wash.
Bake 18-20 min at 425F. Your yolks should be half cooked.
Hollandaise
While your eggs are in the oven, melt butter in small saucepan over low heat, barely to a simmer. Turn off heat.
Separate the yolks into a small bowl.
In a 12oz, wide-mouth mason jar (or the cup that came with blender), add yolks, lemon juice, dijon and salt. Fully seat immersion blender against bottom of jar and turn on low. As you start to see the yolks turn thick, slowly... and I mean slowly... add the butter. Keeping the blades at the bottom until you have a nice emulsification started then move up and down while adding rest of butter.You should have a nice thick but pourable consistency. If you need to thin at all, use a touch of hot water. Taste for seasoning!
Plating
Remove eggs from oven and let cool on countertop, 5 min. Carefully, with a small offset spatula, run around edges of the dough, separating from the tin. Slide spatula underneath and carefully lift from tin.
Plate and drizzle with Hollandaise sauce, garnish with freshly sliced chives and black pepper. Enjoy!
Notes
The Duxelles recipe makes enough for several batches of Egg Wellington (6-8), or you spread it over toast with slices avocado... yum!