First, make the browned butter! (this can be done days in advance). In a medium to large saucepan over medium-low heat, melt the 1 1/4 cup of butter and stir constantly until melted and simmering. Continue to simmer, on medium heat, for at least a few minutes, until the popping and hissing sounds have completely subsided (this is the water in the butter burning off – you’ll know once it’s silent). Continue to cook, stirring constantly, until the butter reaches an amber color and smells nutty or caramelly. The butter will foam up, so keep stirring/lifting up spoonfuls of the butter to see the color. The browned milk solids will foam up in the center of the pan – once it does that- immediately take off the burner and let sit for a few minutes, then pour (scrape ALL of the brown bits – that’s the flavor!) into a heat-proof small bowl and let it cool to room temp (it should be solid). You can refrigerate it now to speed up the cooling.
Then, make the caramel sauce! In a medium saucepan, constantly whisk the white sugar over medium heat until the sugar begins to melt and clump. Continue stirring constantly until sugar has completely melted, then stop stirring and continue to heat (this will be fast!). Once the caramel looks deep gold (not darker) immediately take off heat and vigorously whisk in the room temp butter until it’s fully combined – this may take a minute to bring it all together. Then put back on medium heat and cook for one minute without stirring. Then, whisking constantly, slowly add the heavy cream until fully combined, then stop stirring and let cook for 30 seconds (not longer or it will be too hard!) – it will bubble up.
Finally, remove from heat and whisk in the vanilla and salt, and stir until smooth. Pour into a heatproof container and let sit at room temp to cool. Remember, caramel thickens as it cools so if it seems runny, don't worry.
Then, make your cookies! In a large bowl, using an electric mixer, cream together 1 cup (227g) of the room temp browned butter (it boils down so you should have close to 1 cup left), brown sugar, and white sugar until lighter and fluffy (3-5 minutes – don't skimp on this step!). Add in the egg plus egg yolk and vanilla extract and beat until smooth, another 30 seconds. Add in the flour, starch, baking soda, and salt and mix by hand with a wooden spoon or spatula until fully combined (it will be quite stiff). Scoop dough into a smaller bowl, cover with plastic wrap, and chill in the fridge for 1 hour.
Once the cookie dough is chilled, preheat the oven to 350° Fahrenheit (180° C, no fan) and line a cookie sheet with parchment paper. Lightly grease the parchment and then start scooping out the cookie dough and rolling it into 2-inch round balls. Roll the balls in the crushed cornflakes to coat the entire cookie, then place 4-5 balls on the cookie sheet (~4 inches apart) and bake for 9-10 minutes (do not flatten the balls!) until the edges are just starting to turn golden. Let cookies cool COMPLETELY before removing from the cookie sheet (15 minutes). Finally, drizzle your cooled caramel sauce over the cookies and serve!