Step 1Heat the oven to 300 degrees. Line 2 large baking sheets with parchment paper.
Step 2Spread the pepitas on an unlined rimmed baking sheet. Bake until fragrant, 5 to 7 minutes. Cool completely on the sheet. Raise the oven temperature to 350 degrees.
Step 3Meanwhile, cream the butter, powdered sugar and vanilla together in a large bowl with an electric mixer on medium-high speed until light and fluffy, 3 to 5 minutes. Scrape the bowl and turn the speed to low. Gradually add the flour and beat just until incorporated, then beat in 1 tablespoon milk until smooth.
Step 4Divide the dough into 24 even pieces and roll each into a ball. Arrange the balls 2 inches apart on the prepared baking sheets. Using your thumb or the handle end of a wooden spoon, make a round indentation in the center of each ball. Press the tines of a fork against the edges of each round to imprint decorative lines. Refrigerate until the dough is firm, 15 to 30 minutes.
Step 5Meanwhile, puree the cooled pepitas in a food processor until they become a soft, smooth paste, about 5 minutes. While the pepitas are processing, combine the granulated sugar and 1/4 cup water in a large saucepan. Cook over medium-high heat, swirling the pan occasionally to evenly cook the sugar, until a candy thermometer registers 250 degrees, about 5 minutes. (Tilt the pan if needed for the thermometer to register the temperature.) Remove from the heat and carefully add the pepita paste. Stir until smooth. When the mixture stops steaming, stir in the remaining 3 tablespoons milk. Set aside to cool completely.
Step 6Bake the cookies, one sheet at a time, until golden brown around the edges, 13 to 15 minutes. Cool on the sheets on wire racks. Put a tablespoon of the pepita filling in the thumbprint center of each cookie and spread into an even round. Let stand until the tops of the filling are dry to the touch, about 15 minutes.