Combine coarse sea salt, dried oregano, dried rosemary, dried thyme, rubbed sage, and granulated garlic in a bowl. Rub using your fingers to incorporate. Set aside until ready to use.
Working in batches, pulse mushrooms in a food processor until finely chopped, 6 to 8 pulses. Melt butter in a large skillet over medium-high. Add chopped mushrooms and thyme sprigs; cook, stirring occasionally, until mushrooms begin to brown and stick to skillet, about 20 minutes. Add shallots and garlic; cook, stirring often, until shallots are tender, about 5 minutes. Add Madeira; cook, stirring occasionally, until liquid evaporates and mushrooms begin to stick to bottom of skillet, 2 to 4 minutes. Remove from heat; stir in 1 teaspoon herbed salt. Spread duxelles on a baking sheet; let cool completely, about 30 minutes. Remove and discard thyme sprigs.
Sprinkle beef with remaining 1 teaspoon herbed salt. Heat oil in a large skillet over high. Cook filets until just browned, about 1 minute per side. Transfer to a plate; refrigerate, uncovered, until ready to use.
Place a 12- x 16-inch sheet of plastic wrap on a work surface. Overlap 2 country ham slices to form a cross shape on center of plastic wrap; spread 1/2 cup duxelles evenly over ham. Pat 1 filet dry; brush with 1 1/2 teaspoons mustard, and place in center of cross. Fold ham slices with duxelles over filets, making sure to fully encase beef.
Gather and twist plastic wrap to seal into a tight bundle. Repeat with remaining ham, duxelles, filets, and mustard. Freeze until firm, about 1 hour. (Alternatively, freeze for 3 hours or up to overnight; see step 8.)
Meanwhile, roll 1 puff pastry sheet into a 16- x 12-inch rectangle on a lightly floured surface. Cut out 2 (8-inch) circles. Repeat process with 1 more pastry sheet. (Reserve dough scraps for another use.) Place circles between parchment paper on a baking sheet, and set aside. Roll remaining pastry sheet into a 16-inch square, and cut out 4 (7-inch) circles. Using a square or round cutter, punch out 3/4-inch squares from the 7-inch circles to create a lattice. Discard scraps or reserve for another use. Add lattices to baking sheet; refrigerate until ready to use.
Preheat oven to 425°F. Unwrap 1 filet bundle; center bundle on 1 (8-inch) pastry circle. Lightly brush beaten egg in a 1-inch border around dough edge. Fold dough up and over to encase filet bundle; pinch dough to seal, and trim off any excess dough. Place pastry bundle, seam side down, on parchment paper–lined baking sheet. Repeat process with remaining filet bundles and remaining 8-inch pastry circles. Brush pastry bundles lightly with beaten egg.
Stretch 1 lattice over each pastry bundle. Tuck lattice edges under; trim excess lattice, if needed. Brush lattice lightly with beaten egg; place thyme leaves in each lattice opening. Refrigerate, uncovered, until cold, 10 minutes.
Insert probe of an oven-safe meat thermometer in thickest portion of 1 bundle. (Alternatively, use an instant-read meat thermometer to check temperature during last few minutes of bake time.) Place baking sheet with all 4 bundles in preheated oven; immediately reduce temperature to 400°F. Bake, rotating baking sheet halfway through bake time, until pastry is golden brown and thermometer registers 115°F, 30 to 32 minutes. Let rest on a wire rack 10 minutes before serving. (Beef temperature will rise to 130°F for medium-rare. For rare, freeze bundles in step 2 for 3 hours or up to overnight; bake until thermometer registers 85°F, 35 to 40 minutes. Temperature will rise to 115°F while resting.)
