Prep the grapefruit, which you're going to use three ways: First, use a vegetable peeler to remove 3 or 4 long wide strips of zest, avoiding the bitter white pith underneath. Set zest aside.
Next, take off a thin slice from the top and bottom of grapefruit and upend it on a cut side. Using a thin sharp knife and following the curve of the fruit, cut away peel and pith to expose flesh. Slice off a 2-inch-thick round from one end of grapefruit and squeeze juice from that piece into a small bowl; you'll have about 2 tablespoons. Slice remaining grapefruit crosswise into ½-inch-thick rounds and transfer to a platter. Pick out seeds.
Put ginger into a small mesh sieve set over bowl with grapefruit juice and press down on solids so that ginger juice drips into bowl. Set grapefruit-ginger juice aside (discard solids).
Place shallot slices in a small bowl, cover with cold water, and use your fingertips to separate into individual rings. Soak for 5 minutes, then drain and return to same bowl. Season with salt and add vinegar and 1 teaspoon sesame oil. Set pickled shallots aside.
Season beets with salt. In a large skillet, heat olive oil over medium-high heat until shimmering. Place beets in pan with a flat side down. Cook until almost charred underneath, 6 to 8 minutes, shaking skillet occasionally to make sure they aren't sticking. Turn beets and cook on second flat side until golden brown, 4 to 5 minutes. Cut into one and taste to check doneness- it should be firm at the centre, but no longer completely raw. Transfer to platter with grapefruit and tuck grapefruit slices between and around beet wedges. Season with flaky salt. Let skillet cool for 2 to 3 minutes.
Pour off any oil from skillet and carefully wipe out with paper towel. Return to medium heat, add grapefruit - ginger juice, swirl, and cook just until mixture simmers (this will happen almost immediately).
Add butter and reserved zest and cook until butter is browned and smells nutty, about 4 minutes (if you're using a pan with a dark surface, spoon a little onto a white plate to check colour).
Spoon pan sauce over beets and grapefruit. Top with pickled shallots, along with any juices in bowl. Serve topped with mint.