For the pastry: In the bowl of a stand mixer fitted with a paddle attachment, cream the butter and 1/3 cup of the flour until fluffy. Between two sheets of waxed paper, roll the butter mixture into a 6-by-12-inch rectangle. Place the sheet of butter in the refrigerator or freezer to chill thoroughly.
In the bowl of a stand mixer fitted with a paddle attachment, combine the yeast and 1 1/2 cups of the flour. In a saucepan, heat the milk, sugar and salt over medium heat to 115 to 120 degrees F (just slightly warm). Add it to the flour-yeast mixture. Add the egg and vanilla extract and mix on low speed, scraping the sides constantly, until just combined. Raise the speed to high and beat for 3 minutes. By hand, stir in the remaining 2 cups flour a bit at a time until it forms a medium-soft dough. Place the dough on a floured surface and knead until it feels elastic, about 5 minutes. Cover with plastic wrap and let rest 10 minutes.
Roll out the dough on a floured surface to a 14-inch square. Remove the wax paper from the chilled pad of butter and place the pad on one half of the dough, leaving a 1-inch margin around the sides. Fold the dough over and seal the butter in by pressing the edges of dough together firmly. Roll the rectangle out to a 20-by-12-inch rectangle. Fold it in thirds like you would a business letter. Again roll the rectangle out to a 20-by-12-inch rectangle. Repeat the folding and rolling out twice more, chilling the dough if the butter becomes too soft. Chill the dough for 30 minutes.
Lightly roll out the dough on a floured surface and cut into 4-inch squares. Place them on an ungreased sheet tray and let them rise for 45 minutes.
For the filling: While the dough squares are rising, make the filling. In the bowl of a stand mixer, cream together the goat cheese, cream cheese and sugar until well mixed. Add the salt and egg yolk and mix on high, scraping down the sides once or twice along the way. Set aside in the refrigerator.
Preheat the oven to 425 degrees F.
Slice the pears, skin and all, into thin 1/4-inch slices. Cut these into 1/2-to-1-inch strips. Place in a nonstick skillet over medium heat and add the brandy, honey, cinnamon, nutmeg and cloves and saute until the pears are soft and the sauce around them thickens, about 5 minutes. Remove from the heat and set aside.
For the egg wash: Whisk together the egg and milk. Set aside.
To assemble the Danish: Take about a tablespoon of the cheese filling and place it in the middle of each square, elongating the dollop slightly from one corner to the opposite (do NOT get too close to the edges, the filling spreads as it cooks!). Place a few slices of pear on top of each cheese scoop. Using a pastry brush, lightly brush the margin of exposed dough around the filling with the egg wash. Gently bring the opposite corners (not the ones that the filling is stretching towards) to the middle and seal with a pinch. Once all of the pastries are sealed, give the tops a quick brush of egg wash. Bake until golden brown, 7 to 8 minutes. Serve warm.