Put beets, 4 tbsp. vinegar, 1 tbsp. olive oil, thyme, and bay leaf into a 4-qt. saucepan. Pour in enough water to cover beets by 1" and season with salt and pepper. Bring to a boil and reduce heat to medium-low; simmer until beets are tender and a toothpick slides easily into them, 30–45 minutes. Drain beets and let cool slightly. Working with one beet at a time, put beet between 2 sheets of parchment paper and press with your hand to partially flatten it so that it still remains intact. Repeat with remaining beets and set aside.
Heat 2 tbsp. canola oil in a 12" cast-iron skillet over medium-high heat until almost smoking. Add two smashed beets and cook, flipping once with a metal spatula, until the skins are charred, about 3 minutes. Repeat with remaining canola oil and beets. Season beets with salt and pepper, divide between 4 serving plates, and set aside.
Whisk together remaining vinegar and olive oil in a bowl to make a vinaigrette. Trim and discard thick stems from beet greens and put beet greens and arugula into a medium bowl. Add some of the vinaigrette, toss to combine, and season with salt and pepper. Divide salad between the 4 reserved serving plates; drizzle beets with the remaining vinaigrette and spoon cheese over the beets. Garnish with fried garlic chips, if you like.