Position racks in the middle and upper third of the oven. Heat the oven to 350°F. Line two baking sheets with nonstick liners (like Silpat brand) or with parchment.
Heat the butter, brown sugar, and cream in a medium saucepan over low heat, stirring often, until the butter melts and the brown sugar dissolves. Increase the heat to medium high and, stirring constantly, bring the mixture just to a boil. Remove the pot from the heat and stir in the vanilla, cinnamon, and salt. Stir in the flour until incorporated and then stir in the oatmeal and currants. Drop the batter by the teaspoon 3 inches apart on the baking sheets, about six cookies per baking sheet.
Bake the cookies until evenly light brown, about 10 minutes total. About 5 minutes into baking, switch the sheets from top to bottom and back to front to promote even baking. The cookies won't begin to spread until about 6 minutes into baking.
Line a cooling rack with paper towels. Remove the cookies from the oven and, as soon as they're firm (which will take just a few minutes), use a wide spatula to transfer them to the rack to cool completely. Bake the remaining batter in the same way; the batter will have firmed up a bit, but that's fine.
The cookies will stay crisp and flavorful for a few days wrapped in waxed paper.