Preheat oven to 350 degrees F. Grease and lightly flour or line with paper bake cups twelve 2-1/2-inch muffin cups, four 3-1/2-inch muffin cups, or thirty-six 1-3/4-inch muffin cups. Set aside. In a medium bowl, stir together flour, baking powder, pumpkin pie spice, salt, and baking soda.

In a large bowl, combine eggs, sugar, and cooking oil; beat with an electric mixer on medium speed until combined. Alternately add flour mixture and pumpkin to sugar mixture, beating just until mixture is combined.

Spoon batter into prepared muffin cups, filling ech two-thirds full. Bake until a toothpick inserted in center comes out clean. Allow 20 to 25 minutes for 2-1/2-inch cups, 25 to 30 minutes for 3-1/2-inch cups, or 12 to 15 minutes for the 1-3/4-inch muffin cups. Cool in muffin cups on a wire rack for 5 minutes. Using a knife, loosen edges; carefully remove cakes from muffin cups. Cool completely on rack.

Frost cupcakes with Browned Butter Icing. Arrange shards of Caramelized Hazelnut Brittle or Candied Hazelnuts on top of cupcakes. Makes 12 (2-1/2-inch), 4 (3-1/2-inch), or 36 (1-3/4-inch) cupcakes.

In a small saucepan, heat 3 tablespoons butter over low heat until melted. Continue heating until butter turns a delicate brown. Remove from heat. In a large bowl, combine 1-1/2 cups powdered sugar, 1 tablespoon milk, and 1/2 teaspoon vanilla. Add browned butter. Beat with an electric mixer on low speed until combined. Beat on medium to high speed until it's easy to spread, adding additional milk, if necessary.

Line a large baking sheet with foil; butter foil. Set aside. In a small saucepan, melt 2 teaspoons butter over low heat. Stir in 1/3 cup chopped hazelnuts (filberts); keep hazelnuts warm over low heat. Place 1 cup granulated sugar in a heavy 12-inch skillet; heat over medium-high heat until sugar begins to melt, shaking skillet occasionally to heat sugar evenly. Reduce heat to medium-low; cook until sugar is melted and golden, stirring only as necessary after sugar begins to melt. This should take 12 to 15 minutes. Remove from heat; quickly stir in warm chopped hazelnuts. Immediately pour the mixture onto the prepared baking sheet, allowing syrup to flow and distributing nuts evenly. Cool completely. Break candy into shardlike pieces. Store in a tightly covered container in a cool dry place for up to 1 month.

Using a twisting motion, press a whole hazelnut (filbert) on the pointed end of a 10- to 12-inch wooden skewer. Repeat to skewer a total of 12 hazelnuts. Line a large baking sheet lined with parchment paper. Place sheet on the floor below the kitchen counter. Find something to use as a weight on the blunt ends of the wooden skewers. (Or place a cooling rack over an extra-large bowl, leaving a 2-inch opening on one of the edges. Place the skewers on the cooling rack with the hazelnuts on the edge that is open over the bowl.) Place 1 cup granulated sugar in a heavy 12-inch skillet; heat over medium-high heat until sugar begins to melt, shaking skillet occasionally to heat sugar evenly. Reduce heat to medium-low; cook until sugar is melted and just beginning to turn golden, stirring only as necessary after sugar begins to melt. This should take about 12 minutes. Turn off heat, but leave skillet on the burner. Working quickly, hold the blunt end of a wooden skewer and dip the hazelnut into the sugar syrup, turning to coat well. Place the skewer so the nut hangs over the edge of the counter and the excess syrup drips down onto the parchment-lined baking sheet, making a sugar thread from the nut. Weight the blunt end of the skewer against the counter. (Or place the skewer on rack so the nut hangs over the edge of the rack and the excess sugar drips down into the bowl. Weight the skewer to hold it in place.) Repeat with remaining nuts on skewers, continuing to work quickly. Once all nuts are dipped and sugar has hardened, carefully remove skewers from nuts. Store in a tightly covered container in a cool dry place for up to 1 month.

Prepare as directed through step 3. Place in an airtight container; cover. Store at room temperature for up to 3 days or freeze for up to 1 month. Thaw cakes, if frozen. Continue as directed in step 4.