Arrange racks in upper and lower thirds of oven; preheat to 350°. In a large bowl, using a handheld mixer on medium-high speed, beat butter, brown sugar, and granulated sugar until creamy, 2 to 3 minutes. Add flour and salt and beat until a dough starts to form. Add milk and vanilla and continue to beat until dough comes together.
Using a medium cookie scoop (about 2 Tbsp.), scoop out dough and roll into balls. Arrange between 2 parchment-lined baking sheets, spacing 2" apart. Press a thumbprint into the center of each ball.
Bake cookies, rotating sheets top to bottom halfway through, until golden around the edges and edges are set, 12 to 15 minutes. Using a tablespoon, press indention into cookies again and deepen the center.
In a medium pot over medium heat, cook pecans, brown sugar, corn syrup, and butter, stirring often, until melted and a caramel-like consistency forms, about 5 minutes. Slowly add cream and cook, stirring often, until mixture just starts to simmer, about 2 minutes. Remove from heat and stir in vanilla.
Spoon pecan mixture into center of each cookie. Let sit until cooled, about 15 minutes.
