Cut each beet lengthwise into quarters. If the beets are on the large side, cut them in half, then cut each half into thirds.
Place the beets in an 11-inch sauté pan (with a lid), along with the ginger ale, 1 teaspoon of the salt, and the thyme sprigs. Cover and cook over medium heat for 25-30 minutes, occasionally shaking the pan (gently).
Remove the lid and add the vinegar, honey, the remaining 1/2 teaspoon of salt, black pepper, and cumin. Continue to cook, uncovered, shaking the pan frequently for 7-8 minutes or until the liquid had reduced to a syrup consistency.
Reduce the heat to low then add the butter a tablespoon at a time, swirling the pan to emulsify each dose into the glaze. Finish by stirring in the chopped thyme, then remove from the heat.
Serve immediately with an extra grind or two of black pepper.