Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 10-inch tube pan.
Mix flour, baking soda, salt, cinnamon, and nutmeg together in large bowl.
Beat sugar and eggs with an electric mixer on medium speed in a separate bowl until light and fluffy, about 3 minutes. Beat in oil and 1 teaspoon vanilla extract until well-blended. Mix in flour mixture until batter is smooth. Fold in pears and walnuts. Pour batter into the prepared tube pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 50 to 55 minutes. Cool in the pan for 10 minutes before inverting on a wire rack to cool completely, about 30 minutes.
Melt butter in a saucepan over medium-low heat, stirring constantly until it is a rich brown color and has a nutty aroma, about 5 minutes. Remove from heat; stir in confectioners' sugar and 1 teaspoon vanilla extract until icing is thick and smooth. Stir in water 1 tablespoon at a time to thin icing to desired consistency. Drizzle icing over cooled cake.
