Place 1 finely grated small garlic clove in a medium bowl.
Heat up the butter in a medium skillet over medium-high, swirling often, until melted and foaming. Add the shallot, reduce heat to medium, and cook, stirring occasionally, until the butter turns golden brown, smells nutty, and the shallot is lightly golden (about 3 minutes).
Pour the brown butter mixture into the bowl with the garlic, scraping up browned bits stuck to the bottom of the pan. Add the apple cider vinegar, Dijon mustard, and ½ tsp. salt. Whisk to combine; season with pepper. Set vinaigrette aside.
Heat 1 Tbsp. olive oil in a large skillet over medium-high heat. Add 1 lb. of green beans, sprinkle with 1 tsp salt, and season with freshly ground black pepper. Cook, tossing occasionally, until the beans are bright green and charred in spots but still crisp-tender (about 5–7 minutes). Transfer to a large plate. Repeat the process with the second 1 lb. of green beans, 1 Tbsp. olive oil, 1 tsp. kosher salt, and more pepper.
Heat a medium skillet over medium-high heat. Add the chopped sage leaves to the skillet and drizzle 2 tsp. extra-virgin olive oil over. Sprinkle with ½ tsp salt and season with freshly ground black pepper.
Toast, stirring often, until the sage is crisp, about 3–4 minutes. Transfer the crispy sage to a small bowl. Remove the skillet from heat and return all the green beans to the skillet. Add the reserved vinaigrette and toss to coat. Transfer the dressed green beans to a serving platter and top with the crispy sage.