Heat oven to 350°F and prepare two baking sheets lined with parchment paper.
In the bowl of a stand mixer, beat the butter, sugar, and cheese together until smooth. Add the egg and salt, and beat for 1 minute. Add the flour, one half at a time, and mix until a crumbly dough forms.
Roll the dough into 1-inch balls and place them on the baking sheets, pressing your thumb or index finger into each to make a small indentation (this would be source of the "thumbprint" in the name of the cookie). Fill each indentation with a tiny spoonful of jam and add a pinch of flaky sea salt on top. Chill the cookies for an hour or up to overnight.
Bake the cookies for 12-15 minutes until done and let cool on a rack before storing in an airtight container. Serve these savory-sweet delights as an accompaniment to a charcuterie platter or just as an upgrade to your usual cookie game!