Trim and quarter mushrooms; place on 17- x 11-inch (45 x 29 cm) rimmed baking sheet.
Sprinkle with vinegar and half each of the salt and pepper; toss to coat mushrooms evenly.
Bake in 400°F (200°C) oven, stirring occasionally, for 35 to 40 minutes or until tender and liquid has evaporated. Let cool slightly.
Drain red peppers; pat dry with paper towels.
Chop and place in large bowl.
Add half of the bread crumbs, the honey, mint and remaining salt and pepper; toss to combine.
Cut rind from cheese; shred and add to bowl along with mushrooms.
Toss again; set aside. (Filling can be covered and refrigerated for up to 1 day; stir before using.)
Blend mustard with 1 tablespoon (15 mL) water; set aside.
Layer 3 sheets of phyllo on work surface; cover remaining phyllo with waxed paper or plastic wrap, then damp towel, to prevent drying out.
Using sharp knife and 8-inch (20 cm) round cake pan or cardboard template as guide, cut out 2 circles from each sheet, discarding trimmings.
Place 2 circles side by side on work surface. Brush each lightly with mustard mixture.
Top each with second circle, brush lightly with butter. Top each with remaining circles; brush again with mustard.
Sprinkle 1 tablespoon (15 mL) of the breadcrumbs in centre of each. Top crumbs with heaping 1/2 cup (125 mL) of the filling.
Fold 1 edge of phyllo to centre over filling.
Grasp point at edge of circle where first fold begins; fold point over filling into centre, forming pleat.
Continue folding phyllo over filling, overlapping pleats but leaving 1/2-inch (1 cm) diameter steam vent in centre; press to seal.
Place on parchment paper lined or greased baking sheet.
Brush all over with egg white.
Repeat to make 8 puffs. (Puffs can be refrigerated for up to 12 hours. Or freeze, then wrap each in plastic wrap and foil and store in airtight container for up to 2 weeks; increase baking by 5 minutes).
Bake in 400°F (200°C) oven for 20 minutes or until golden brown and crispy.