Grillpølsen er gennemstegt.

Because Danish normally writes compound nouns as a single word.

Here, grill + pølse becomes grillpølse. English often uses separate words where Danish prefers one compound word. Then the definite ending is added to the whole compound, giving grillpølsen.

In Danish, the definite article is usually attached to the noun itself.

So:

• en pølse = a sausage
• pølsen = the sausage

Because pølse ends in an unstressed -e, the definite form is written with -n rather than a full extra -en. That is why grillpølsen already means the grill sausage.

A very useful extra point: if there is an adjective before a definite noun, Danish usually also adds a separate article:

• den gennemstegte grillpølse

That depends on the noun’s grammatical gender.

pølse is a common gender noun:

• en pølse
• pølsen

A neuter noun would be:

• et æg
• ægget

In Danish compounds, the last part usually decides the gender. Since pølse is common gender, grillpølse is also common gender, so its definite form is grillpølsen.

er is the present-tense form of være = to be.

So Grillpølsen er gennemstegt describes a present state: the sausage is cooked through.

It is different from bliver:

• Grillpølsen er gennemstegt = it is cooked through
• Grillpølsen bliver gennemstegt = it is getting cooked through / it becomes cooked through

Also, Danish present tense does not change for person:

• jeg er
• du er
• han er
• de er

Here gennemstegt is a participle used like an adjective.

It is built from:

• gennem- = through
• stegt = fried/roasted/cooked

The -t is part of the normal form stegt here. It is not an extra ending added just because grillpølsen is singular.

That is why you also say:

• en gennemstegt pølse

When this adjective takes an -e form, you get:

• den gennemstegte pølse
• pølserne er gennemstegte

Yes.

In your sentence, gennemstegt comes after er, so it is a predicative adjective:

• Grillpølsen er gennemstegt

Before the noun, it becomes an attributive adjective:

• en gennemstegt grillpølse

If the noun is definite, Danish normally uses:

• den gennemstegte grillpølse

That pattern is often called double definiteness: a separate article plus a definite-style adjective, while the noun itself stays without the attached definite ending.

For a yes/no question, Danish usually puts the finite verb first:

• Er grillpølsen gennemstegt?

So the statement is:

• Grillpølsen er gennemstegt.

And the question is:

• Er grillpølsen gennemstegt?

More generally, Danish has a strong verb-second pattern in statements. If another element comes first, the verb still stays in second position:

• Nu er grillpølsen gennemstegt.

There is no exact English equivalent for ø.

A useful approximation:

• round your lips as if you were saying oo
• but try to say eh

If you know German or French, it is close to:

• German ö
• French eu

A couple of extra pronunciation tips:

• grillpølsen normally has main stress on grill
• the -e in pølse is weak
• the final -n in pølsen is light

The best way to learn ø is to hear it from native speakers and imitate it, because English does not have the same vowel.