Prepare ingredients: peel and thinly slice the onions, mince the garlic, grate the cheddar and measure the milk and stock. Having everything prepped keeps the cooking flow smooth and prevents overcooking when you move between steps.

Caramelize the onions: in a large heavy-bottomed pot or Dutch oven, heat the olive oil and 1 tablespoon of butter over medium-low heat. Add the sliced onions and 1/2 teaspoon of the salt. Cook gently, stirring every 5–7 minutes, for 30–40 minutes until deeply golden and jammy. Lower, slow caramelization converts the onions' starches to sugars and builds the soup’s sweet, savory backbone—rushing this step yields bitter or pale onions.

Add garlic near the end of caramelizing: when the onions are soft and starting to color (after ~20–25 minutes), push them to the side and add the remaining 1 tbsp butter and the minced garlic. Sauté 1–2 minutes until fragrant but not browned. Garlic burns quickly; adding it late preserves its aromatic, bright flavor.

Build a light roux: raise heat to medium and sprinkle the flour over the onion–garlic mixture. Stir constantly for 1–2 minutes to cook the raw flour taste and to coat the vegetables. This roux will thicken the milk without needing heavy cream; cooking the flour briefly prevents a pasty flavor.

Deglaze and gently temper the dairy: pour in the stock while stirring to lift any browned bits (those carry concentrated flavor). Then slowly whisk in the milk in two additions, whisking constantly to avoid lumps. Bring to a gentle simmer—don’t boil hard, which can cause milk proteins to separate. Tempering and gradual heat builds a smooth, stable emulsion.

Simmer to marry flavors and thicken: reduce the heat to low and simmer, uncovered, for 12–15 minutes until the soup is slightly thickened and coats the back of a spoon. Taste and season with the remaining salt and pepper and the thyme. A gentle simmer cooks out any raw flour and allows the onion sweetness to meld with the dairy.

Finish with cheese off the heat: remove the pot from the burner and stir in three-quarters of the grated cheddar until fully melted and glossy. Adding cheese off the heat (or on the lowest heat) prevents the cheese from seizing and becoming grainy; slow incorporation yields a silky texture.

Adjust texture: for a rustic-textured soup leave as-is, or use an immersion blender to purée until silky. If you prefer an ultra-smooth restaurant finish, strain the soup through a coarse sieve, pressing with a spatula to extract flavor. Blending rounds out the texture and suspends the cheese uniformly.

Brighten and balance: stir in the optional teaspoon of lemon juice or vinegar to lift the richness. Taste and correct salt and pepper. Bright acid sharpens the cheddar and prevents the soup from feeling cloying.

Finish and serve: ladle the soup into warmed bowls and sprinkle the remaining grated cheddar over each serving. For a restaurant touch, drizzle a little good olive oil and finish with a few thyme leaves or cracked black pepper. Serve with crusty bread if desired.