Preheat oven to 400°F. Wrap each beet in aluminum foil. Roast until very tender, 1 hour and 15 minutes to 1 hour and 30 minutes.
Remove beets from oven. Open foil packets slightly to vent and let stand until cool enough to handle, about 10 minutes. Using a clean towel, gently rub peels off beets; discard peels. Slice beets crosswise into 1/2-inch-thick rounds.
Melt 1 tablespoon of the butter in a skillet over medium-high heat. Add mushrooms, shallot, and 1 tablespoon of the thyme. Cook, stirring occasionally, until mushrooms are browned, 6 to 8 minutes. Add wine and bring to a boil. Cook until liquid has almost evaporated, about 3 minutes. Add broth and bring to a boil. Cook until liquid is reduced by half, 5 to 6 minutes. Remove from heat. Stir in mustard, salt, pepper, and remaining 1 tablespoon butter until butter is melted and incorporated.
Fan out beet slices on each of 2 plates. Top evenly with mushroom bordelaise and sprinkle with remaining 1/2 tablespoon thyme.
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