Set your oven to 375 degrees Fahrenheit and position the rack to the middle position. Remove your pie crust from the refrigerator so it can come to room temperature while you prep the filling.

Wash each pear under cool water and pat dry. Using a sharp knife, peel the skin away, halve the pear lengthwise, and scoop out the core with a melon baller or small spoon, then cut each half into half-inch slices.

In a large bowl, whisk together the granulated sugar, brown sugar, cornstarch, cardamom, and salt.

Add the sliced pears to the sugar mixture along with the vanilla extract, lemon juice, and lemon zest. Gently fold everything together using a silicone spatula, being careful not to break the pear slices. Let this sit for 2 minutes.

Gently unroll your pie crust and fit it into the 9-inch pie dish, pressing it lightly into the bottom and sides. Trim any overhang to about an inch beyond the rim, then fold that edge under itself and crimp with your fingers to create a finished edge.

Pour the pear mixture into the crust, mounding it slightly in the center. Scatter the butter pieces over the top of the filling.

Beat the egg in a small bowl and brush it lightly over the crust edges and top. Sprinkle coarse sugar over the egg wash if desired.

Place the pie in the preheated 375-degree oven and bake for 45 to 50 minutes, until the crust is deep golden brown and you see the filling starting to bubble at the edges. If the crust edges brown too quickly, loosely tent them with foil after 25 minutes.

Remove the pie from the oven and place it on a wire cooling rack. Let it cool at room temperature for at least 3 hours before slicing to allow the filling to set properly.