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Heat oil (or melt butter) in a large stockpot over medium-high heat. Add onion and sauté for 3 minutes, stirring occasionally. Stir in the celery, carrots and garlic. Continue sautéing, for 4 more minutes, stirring occasionally. Stir in the flour until combined, and sauté for 1 more minute, stirring occasionally.
Gradually stir in the chicken stock until it is completely combined. Stir in the milk, potatoes, ham, bay leaf, dried thyme, salt and pepper until combined. Continue cooking until the mixture reaches a simmer, but not boiling.
Reduce heat to medium-low, cover, and simmer for about 10-15 minutes or until the potatoes are soft, stirring occasionally every few minutes so that the bottom of the soup does not burn. (The smaller you dice your potatoes, the faster they will cook.)
Once the potatoes are soft, give the soup a taste and season with extra salt and pepper if needed. Serve immediately.