Preheat oven to 350°F. Grease or lightly spray a standard 12-cup muffin tin, or use paper liners.
In a small mixing bowl, shift together 1 cup of flour, cardamom, baking soda, and salt. Fold in the ½ cup ground walnuts and set aside.
Toss the diced pear with the remaining 2 tablespoons of flour, making sure all pieces are well covered. Set aside.
In a two cup measure, combine the buttermilk and vanilla. Set aside.
Using a stand mixer fitted with the paddle attachment, combine the sugars and butter on medium speed until combined, about 3-4 minutes. Add the egg and blend just until incorporated. Turn the mixer down to a low speed and add one-third of the flour mixture and one-third of the buttermilk. Scrape down the sides of the bowl as needed and beat just until combined. Add the remaining flour and buttermilk mixture, in two additions. Scrape down the sides and using a rubber spatula, fold the batter over a few times to ensure it is well mixed.
Add the diced and floured pears and fold together using the spatula.
Using an ice cream scoop, divide the batter between 12 muffin cups, filling them three-quarters of the way. Sprinkle the muffins with the turbinado sugar and the chopped walnuts if desired. Bake until the muffins are firm on the top, and golden brown, about 25 minutes.
Allow the muffins to cool in the pan, about 10 minutes. Cool completely on a wire rack or serve slightly warm.
Store leftovers in an airtight container at room temperature, up to three days.