Whisk in the eggs one at a time, beating well after each addition until the batter becomes more fluid and satin-like. Add the pumpkin purée and vanilla, then whisk again until fully integrated. The batter should look smooth, glossy, and evenly colored, with no streaks of pumpkin remaining. This is the key emulsification stage: you are marrying fat, sugar, egg, and moisture into one stable base. Critical point: whisk thoroughly, but do not whip aggressively for several minutes as though making sponge cake. Too much air will push the brownies toward a cakier texture instead of the intended dense, luxurious crumb.