In a medium saucepan over medium-high heat, combine the sugar and corn syrup and bring to a boil. Cook, without stirring, for 3 minutes. Stir to dissolve the remaining sugar, then boil for another 2 minutes, stirring occasionally.
Add the butter, cinnamon, cayenne, salt, and pecans and stir to incorporate. Let boil for another 2 minutes, stirring occasionally.
Remove the pan from the heat and add the rum extract and baking soda, stirring to combine.
Quickly pour the mixture onto the prepared baking sheet and let cool completely, 10–15 minutes.
Break the brittle into pieces. Store in an airtight container at room temperature for up to one week.