In a large bowl, combine the ground venison and mayonnaise. Mix well until incorporated. Add the chopped apricots, grated garlic, sage, smoked paprika, kosher salt, and black pepper. Mix gently until just combined. Shape into 4 oval patties, each about ¾-inch thick. Place patties on a plate or tray and chill for 10 minutes to firm. Cook a small test piece in a skillet to check seasoning. Adjust as needed.

Heat neutral oil in a skillet over medium-high heat. Add patties and sear for 3–4 minutes on one side until golden brown. You'll know it's ready to flip once it releases from the pan. It sticks because the crust is still forming, so don't nudge it around if it's not releasing. The protein will almost always tell you when it's ready. Flip, reduce heat to medium, and add butter, smashed garlic cloves, and thyme sprigs to the skillet. Tilt the pan and baste continuously with the foaming butter for 2–3 minutes until the patties are fully cooked through and glazed with herb butter.

In a mixing bowl, toss arugula with lemon juice, extra virgin olive oil, and a pinch of flaky salt. Just before serving, top with shaved Pecorino Romano.