Cut beef into 1 inch cubes. Season with 1 teaspoon salt, then arrange in an even layer on a baking sheet; freeze until solid, at least 1 hour
Heat butter in a large nonstick or cast-iron skillet over medium-low.
For the duxelle, pulse shallot and garlic in a mini food processor until finely minced, about 30 seconds, scraping down slides of bowl as needed. Transfer to skillet with butter and cook over medium low, stirring frequently, until shallots have softened, about 3 minutes.
Meanwhile, add half of mushrooms to food processor and pulse until finely minced, about 15 one-second pulses, scraping down sides of bowl as needed. Transfer to skillet; repeat with remaining mushrooms.
Add remaining minced mushrooms, ¼ teaspoon salt and ¼ teaspoon pepper to skillet with shallots and cook over medium heat, stirring occasionally, until moisture has evaporated and mushrooms are softened and barely starting to brown, 17–20 minutes. Add red wine and cook, stirring frequently, until moisture has evaporated off, about 2 minutes. Off heat, stir in sage. Transfer duxelle to a bowl and let cool.
Mix together Boursin and Dijon.
Heat oven to 425ºF with rack set in middle position.
Unfold pastry, cut each pastry square into three (9½✕3-inch) strips, along existing seems.
Working with one strip at a time, use a rolling pin to roll out to a 15✕3-inch strip.
Cut into 5 (3X3-inch) squares.
Repeat rolling and cutting with remaining strips.
Scoop 1 teaspoon duxelle onto bottom third of each pastry square, leaving ¼ inch border at bottom. Scoop ½ teaspoon cheese mixture onto pastry, just above duxelle. (Reserve any remaining cheese mixture for serving.)
Place 1 piece frozen beef in between duxelle and cheese mixture.
For the egg wash, beat egg and 1 teaspoon water.
Brush egg wash around left side, right side, and top edge. Fold pastry over filling, toward you, like a book.
Press to seal then using a fork, crimp edges. Trim off any excess dough.
Brush bites with egg wash and bake until puffed and golden brown, 10–12 minutes.
Whisk cream, half-and-half or milk by tablespoon into reserved cheese mixture until desired consistency is met. Serve alongside wellington bites as a dipping sauce.