Cook butter and lard in medium saucepan over medium heat, stirring often, until it foams, then browns, 5-8 minutes. If not using butter just melt leaf lard. Scrape into a large bowl and let cool slightly.
Whisk flour, baking soda, and kosher salt in a medium bowl.
Add brown sugar and granulated sugar to browned butter. Using an electric mixer on medium speed, beat until incorporated, about 1 minutes. Add eggs and vanilla and beat until mixture lightens and begins to thicken, about 30 seconds. Reduce mixer speed to low; add dry ingredients and beat just to combine. Mix in toffee piece and chocolate chips with a wooden spoon or rubber spatula.
Let dough sit at room temperature at least 30 minutes to allow the flour to hydrate. Dough will look very loose at first but will thicken as it sits.
Place a rack in the middle of oven; preheat to 375 degrees.
Portion out balls of dough and place on a parchment-lined baking sheet, spacing about 3 inches apart. Do not flatten; cookies will spread as they bake. Sprinkle with sea salt.
Bake cookies until edges are golden brown and firm but centers are still soft, 9-11 minutes.
Let cool on baking sheet 10 minutes, then transfer to a wire rack and let cool completely.
Repeat with remaining dough and another parchment-lined baking sheet to make 10 more cookies.
Note: can be made 1 day ahead of time, chilled, and brought to room temperature before baking.