Your question looks at the D&D 5E ability check system a little bit backwards. The mechanic is in fact called ability checks, not skill checks or tool checks, and it's intended to be used looking at the ability first, and then figuring out whether proficiency in a skill or tool might apply. From the Ability Checks section of chapter 7 (my emphasis added):
For every ability check, the DM decides which of the six abilities is relevant to the task at hand and the difficulty of the task, represented by a Difficulty Class.
And then looking at how Tools work from Chapter 5:
Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use. For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver's tools, or a Strength check to make something out of particularly hard wood.
So rather than looking at it from the perspective of "How do I do a check with cooking utensils?", we need to look at it from the perspective of "Why is the DM calling for an ability check?" There are plenty of things involving cooking utensils that probably wouldn't call for a check at all, since making a normal everyday meal while in a home with all the ingredients handy isn't really something that tends to have much of a chance of failure or much impact of success. But sometimes the character is trying to do something where it makes sense to use a roll to determine if it succeeds, so the DM would call for an ability check. And then if having proficiency with cooking utensils would be helpful to accomplishing the objective, then the proficiency bonus would be added. Here are some ability checks I can imagine calling for where one could reasonably add the proficiency bonus for cooking utensils:
- A Strength check to cut open a large animal that requires a lot of effort, but you need to know how to cut it properly to in order to prepare it correctly. (Thanks ZwiQ for this suggestion.)
- A Dexterity check to make something with fine detail, like decorating a cake.
- A Constitution check to produce food taking a long time in a hot, muggy, kitchen full of fumes.
- An Intelligence check to remember an old family recipe.
- A Wisdom check to observe how somebody else is making a recipe to be able to copy it, or to decipher what kind of food they might like.
- A Charisma check to make food in an appealing way, or to try to make friends using food.
As you've noticed, other than the general suggestions provided in the "Using Each Ability" section, it's really completely up to the DM when ability checks are called for, and which ability applies to the task that the character is trying to do. Different DMs definitely have different philosophies and approaches to it. And sometimes really more than one skill or tool proficiency could reasonably apply. I've certainly (as DM) called for Charisma checks allowing for either their Persuasion or Cooking Utensils proficiency to be applied, whichever they had, since they were trying to make friends using food. But once the DM picks which ability to use for the ability check for whatever it is that the character is trying to do that requires a roll, it's not usually that hard to figure out if proficiency in a tool might apply to the roll.
Note that since "Your proficiency bonus can’t be added to a single die roll or other number more than once," (Ch. 7, "Proficiency Bonus") even if you have proficiency in both a skill and a tool that might apply to the ability check, you still only add your proficiency bonus once. (And Blckknght in the comments says that there is an optional rule suggested in Xanathar's Guide that could grant advantage in that situation, but I don't have that book so you'll have to look that up for yourself.)