Heat a large, heavy skillet over medium-high heat. Add 1 tablespoon oil to pan; swirl to coat.
Add half of the beef. Cook 8 minutes or until well browned, turning to brown on all sides.
Place browned beef in a 6-quart electric slow cooker; sprinkle with ⅛ teaspoon salt.
Repeat procedure with the remaining 1 tablespoon oil, remaining beef, and ⅛ teaspoon salt.
Add onions to pan; cook 1 minute on each side or until browned.
Add onions to slow cooker. Add garlic to pan; sauté 1 minute. Add tomato paste; cook 1 minute, stirring constantly.
Add ½ cup wine, scraping pan to loosen browned bits; cook 2 minutes or until half of liquid evaporates.
Add stock and pepper; bring to a simmer.
Carefully pour stock mixture into slow cooker; nestle thyme sprigs into liquid.
Top beef mixture with mushrooms; scatter carrots over top.
Cover and cook on LOW for 7½ hours.
Strain mixture through a colander over a Dutch oven or large skillet; discard thyme sprigs.
Bring cooking liquid to a boil; cook 6 minutes or until liquid is reduced to 2 cups.
Stir in remaining ¾ teaspoon salt. Combine remaining 2 tablespoons wine and flour, stirring with a whisk to form a slurry.
Stir into cooking liquid; cook 1 minute or until bubbly and thickened, stirring constantly.
Pour over beef mixture; toss gently to combine. Garnish with thyme sprigs.